measure_theory.integral.average | Mathlib porting status

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feat(measure_theory/integral/average): Lebesgue average (#19199)

Define the Lebesgue integral version of the average of a measurable function and prove the corresponding first moment method.

Diff

@@ -19,6 +19,8 @@ measure, then the average of any function is equal to its integral.
 For the average on a set, we use `⨍ x in s, f x ∂μ` (notation for `⨍ x, f x ∂(μ.restrict s)`). For
 average w.r.t. the volume, one can omit `∂volume`.
 
+Both have a version for the Lebesgue integral rather than Bochner.
+
 We prove several version of the first moment method: An integrable function is below/above its
 average on a set of positive measure.
 
@@ -28,11 +30,6 @@ The average is defined as an integral over `(μ univ)⁻¹ • μ` so that all t
 integrals work for the average without modifications. For theorems that require integrability of a
 function, we provide a convenience lemma `measure_theory.integrable.to_average`.
 
-## TODO
-
-Provide the first moment method for the Lebesgue integral as well. A draft is available on branch
-`first_moment_lintegral`.
-
 ## Tags
 
 integral, center mass, average value
@@ -44,19 +41,174 @@ open_locale topology big_operators ennreal convex
 variables {α E F : Type*} {m0 : measurable_space α}
   [normed_add_comm_group E] [normed_space ℝ E] [complete_space E]
   [normed_add_comm_group F] [normed_space ℝ F] [complete_space F]
-  {μ : measure α} {s t : set α}
+  {μ ν : measure α} {s t : set α}
 
 /-!
 ### Average value of a function w.r.t. a measure
 
-The average value of a function `f` w.r.t. a measure `μ` (notation: `⨍ x, f x ∂μ`) is defined as
-`(μ univ).to_real⁻¹ • ∫ x, f x ∂μ`, so it is equal to zero if `f` is not integrable or if `μ` is an
-infinite measure. If `μ` is a probability measure, then the average of any function is equal to its
-integral.
-
+The (Bochner, Lebesgue) average value of a function `f` w.r.t. a measure `μ` (notation:
+`⨍ x, f x ∂μ`, `⨍⁻ x, f x ∂μ`) is defined as the (Bochner, Lebesgue) integral divided by the total
+measure, so it is equal to zero if `μ` is an infinite measure, and (typically) equal to infinity if
+`f` is not integrable. If `μ` is a probability measure, then the average of any function is equal to
+its integral.
 -/
 
 namespace measure_theory
+section ennreal
+variables (μ) {f g : α → ℝ≥0∞}
+include m0
+
+/-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. a measure `μ`, notation: `⨍⁻ x, f x ∂μ`.
+It is defined as `μ univ⁻¹ * ∫⁻ x, f x ∂μ`, so it is equal to zero if `μ` is an infinite measure. If
+`μ` is a probability measure, then the average of any function is equal to its integral.
+
+For the average on a set, use `⨍⁻ x in s, f x ∂μ` (defined as `⨍⁻ x, f x ∂(μ.restrict s)`). For
+average w.r.t. the volume, one can omit `∂volume`. -/
+noncomputable def laverage (f : α → ℝ≥0∞) := ∫⁻ x, f x ∂((μ univ)⁻¹ • μ)
+
+notation `⨍⁻` binders `, ` r:(scoped:60 f, f) ` ∂` μ:70 := laverage μ r
+notation `⨍⁻` binders `, ` r:(scoped:60 f, laverage volume f) := r
+notation `⨍⁻` binders ` in ` s `, ` r:(scoped:60 f, f) ` ∂` μ:70 :=
+  laverage (measure.restrict μ s) r
+notation `⨍⁻` binders ` in ` s `, ` r:(scoped:60 f, laverage (measure.restrict volume s) f) := r
+
+@[simp] lemma laverage_zero : ⨍⁻ x, (0 : ℝ≥0∞) ∂μ = 0 := by rw [laverage, lintegral_zero]
+
+@[simp] lemma laverage_zero_measure (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂(0 : measure α) = 0 :=
+by simp [laverage]
+
+lemma laverage_eq' (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂((μ univ)⁻¹ • μ) := rfl
+
+lemma laverage_eq (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ / μ univ :=
+by rw [laverage_eq', lintegral_smul_measure, ennreal.div_eq_inv_mul]
+
+lemma laverage_eq_lintegral [is_probability_measure μ] (f : α → ℝ≥0∞) :
+  ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ :=
+by rw [laverage, measure_univ, inv_one, one_smul]
+
+@[simp] lemma measure_mul_laverage [is_finite_measure μ] (f : α → ℝ≥0∞) :
+  μ univ * ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ :=
+begin
+  cases eq_or_ne μ 0 with hμ hμ,
+  { rw [hμ, lintegral_zero_measure, laverage_zero_measure, mul_zero] },
+  { rw [laverage_eq, ennreal.mul_div_cancel' (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)] }
+end
+
+lemma set_laverage_eq (f : α → ℝ≥0∞) (s : set α) : ⨍⁻ x in s, f x ∂μ = ∫⁻ x in s, f x ∂μ / μ s :=
+by rw [laverage_eq, restrict_apply_univ]
+
+lemma set_laverage_eq' (f : α → ℝ≥0∞) (s : set α) :
+  ⨍⁻ x in s, f x ∂μ = ∫⁻ x, f x ∂((μ s)⁻¹ • μ.restrict s) :=
+by simp only [laverage_eq', restrict_apply_univ]
+
+variable {μ}
+
+lemma laverage_congr {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : ⨍⁻ x, f x ∂μ = ⨍⁻ x, g x ∂μ :=
+by simp only [laverage_eq, lintegral_congr_ae h]
+
+lemma set_laverage_congr (h : s =ᵐ[μ] t) : ⨍⁻ x in s, f x ∂μ = ⨍⁻ x in t, f x ∂μ :=
+by simp only [set_laverage_eq, set_lintegral_congr h, measure_congr h]
+
+lemma set_laverage_congr_fun (hs : measurable_set s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
+  ⨍⁻ x in s, f x ∂μ = ⨍⁻ x in s, g x ∂μ :=
+by simp only [laverage_eq, set_lintegral_congr_fun hs h]
+
+lemma laverage_lt_top (hf : ∫⁻ x, f x ∂μ ≠ ∞) : ⨍⁻ x, f x ∂μ < ∞ :=
+begin
+  obtain rfl | hμ := eq_or_ne μ 0,
+  { simp },
+  { rw laverage_eq,
+    exact div_lt_top hf (measure_univ_ne_zero.2 hμ) }
+end
+
+lemma set_laverage_lt_top : ∫⁻ x in s, f x ∂μ ≠ ∞ → ⨍⁻ x in s, f x ∂μ < ∞ := laverage_lt_top
+
+lemma laverage_add_measure :
+  ⨍⁻ x, f x ∂(μ + ν) =
+    μ univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂μ + ν univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂ν :=
+begin
+  by_cases hμ : is_finite_measure μ, swap,
+  { rw not_is_finite_measure_iff at hμ,
+    simp [laverage_eq, hμ] },
+  by_cases hν : is_finite_measure ν, swap,
+  { rw not_is_finite_measure_iff at hν,
+    simp [laverage_eq, hν] },
+  haveI := hμ, haveI := hν,
+  simp only [←ennreal.mul_div_right_comm, measure_mul_laverage, ←ennreal.add_div,
+    ←lintegral_add_measure, ←measure.add_apply, ←laverage_eq],
+end
+
+lemma measure_mul_set_laverage (f : α → ℝ≥0∞) (h : μ s ≠ ∞) :
+  μ s * ⨍⁻ x in s, f x ∂μ = ∫⁻ x in s, f x ∂μ :=
+by { haveI := fact.mk h.lt_top, rw [← measure_mul_laverage, restrict_apply_univ] }
+
+lemma laverage_union (hd : ae_disjoint μ s t) (ht : null_measurable_set t μ) :
+  ⨍⁻ x in s ∪ t, f x ∂μ =
+    μ s / (μ s + μ t) * ⨍⁻ x in s, f x ∂μ + μ t / (μ s + μ t) * ⨍⁻ x in t, f x ∂μ :=
+by rw [restrict_union₀ hd ht, laverage_add_measure, restrict_apply_univ, restrict_apply_univ]
+
+lemma laverage_union_mem_open_segment (hd : ae_disjoint μ s t) (ht : null_measurable_set t μ)
+  (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) :
+  ⨍⁻ x in s ∪ t, f x ∂μ ∈ open_segment ℝ≥0∞ (⨍⁻ x in s, f x ∂μ) (⨍⁻ x in t, f x ∂μ) :=
+begin
+  refine ⟨μ s / (μ s + μ t), μ t / (μ s + μ t), ennreal.div_pos hs₀ $ add_ne_top.2 ⟨hsμ, htμ⟩,
+    ennreal.div_pos ht₀ $ add_ne_top.2 ⟨hsμ, htμ⟩, _, (laverage_union hd ht).symm⟩,
+  rw [←ennreal.add_div, ennreal.div_self (add_eq_zero.not.2 $ λ h, hs₀ h.1)
+    (add_ne_top.2 ⟨hsμ, htμ⟩)],
+end
+
+lemma laverage_union_mem_segment (hd : ae_disjoint μ s t) (ht : null_measurable_set t μ)
+  (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) :
+  ⨍⁻ x in s ∪ t, f x ∂μ ∈ [⨍⁻ x in s, f x ∂μ -[ℝ≥0∞] ⨍⁻ x in t, f x ∂μ] :=
+begin
+  by_cases hs₀ : μ s = 0,
+  { rw ← ae_eq_empty at hs₀,
+    rw [restrict_congr_set (hs₀.union eventually_eq.rfl), empty_union],
+    exact right_mem_segment _ _ _ },
+  { refine ⟨μ s / (μ s + μ t), μ t / (μ s + μ t), zero_le _, zero_le _, _,
+      (laverage_union hd ht).symm⟩,
+    rw [←ennreal.add_div, ennreal.div_self (add_eq_zero.not.2 $ λ h, hs₀ h.1)
+      (add_ne_top.2 ⟨hsμ, htμ⟩)] }
+end
+
+lemma laverage_mem_open_segment_compl_self [is_finite_measure μ] (hs : null_measurable_set s μ)
+  (hs₀ : μ s ≠ 0) (hsc₀ : μ sᶜ ≠ 0) :
+  ⨍⁻ x, f x ∂μ ∈ open_segment ℝ≥0∞ (⨍⁻ x in s, f x ∂μ) (⨍⁻ x in sᶜ, f x ∂μ) :=
+by simpa only [union_compl_self, restrict_univ]
+  using laverage_union_mem_open_segment ae_disjoint_compl_right hs.compl hs₀ hsc₀
+    (measure_ne_top _ _) (measure_ne_top _ _)
+
+@[simp] lemma laverage_const (μ : measure α) [is_finite_measure μ] [h : μ.ae.ne_bot] (c : ℝ≥0∞) :
+  ⨍⁻ x, c ∂μ = c :=
+by simp only [laverage_eq, lintegral_const, measure.restrict_apply, measurable_set.univ, univ_inter,
+  div_eq_mul_inv, mul_assoc, ennreal.mul_inv_cancel, mul_one, measure_ne_top μ univ, ne.def,
+  measure_univ_ne_zero, ae_ne_bot.1 h, not_false_iff]
+
+lemma set_laverage_const (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : ℝ≥0∞) : ⨍⁻ x in s, c ∂μ = c :=
+by simp only [set_laverage_eq, lintegral_const, measure.restrict_apply, measurable_set.univ,
+  univ_inter, div_eq_mul_inv, mul_assoc, ennreal.mul_inv_cancel hs₀ hs, mul_one]
+
+@[simp] lemma laverage_one [is_finite_measure μ] [h : μ.ae.ne_bot] : ⨍⁻ x, (1 : ℝ≥0∞) ∂μ = 1 :=
+laverage_const _ _
+
+lemma set_laverage_one (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) : ⨍⁻ x in s, (1 : ℝ≥0∞) ∂μ = 1 :=
+set_laverage_const hs₀ hs _
+
+@[simp] lemma lintegral_laverage (μ : measure α) [is_finite_measure μ] (f : α → ℝ≥0∞) :
+  ∫⁻ x, ⨍⁻ a, f a ∂μ ∂μ = ∫⁻ x, f x ∂μ :=
+begin
+  unfreezingI { obtain rfl | hμ := eq_or_ne μ 0 },
+  { simp },
+  { rw [lintegral_const, laverage_eq,
+      ennreal.div_mul_cancel (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)] }
+end
+
+lemma set_lintegral_set_laverage (μ : measure α) [is_finite_measure μ] (f : α → ℝ≥0∞) (s : set α) :
+  ∫⁻ x in s, ⨍⁻ a in s, f a ∂μ ∂μ = ∫⁻ x in s, f x ∂μ :=
+lintegral_laverage _ _
+
+end ennreal
+
 section normed_add_comm_group
 variables (μ) {f g : α → E}
 include m0
@@ -114,9 +266,13 @@ variable {μ}
 lemma average_congr {f g : α → E} (h : f =ᵐ[μ] g) : ⨍ x, f x ∂μ = ⨍ x, g x ∂μ :=
 by simp only [average_eq, integral_congr_ae h]
 
-lemma set_average_congr_set_ae (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ :=
+lemma set_average_congr (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ :=
 by simp only [set_average_eq, set_integral_congr_set_ae h, measure_congr h]
 
+lemma set_average_congr_fun (hs : measurable_set s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
+  ⨍ x in s, f x ∂μ = ⨍ x in s, g x ∂μ :=
+by simp only [average_eq, set_integral_congr_ae hs h]
+
 lemma average_add_measure [is_finite_measure μ] {ν : measure α} [is_finite_measure ν] {f : α → E}
   (hμ : integrable f μ) (hν : integrable f ν) :
   ⨍ x, f x ∂(μ + ν) =
@@ -180,7 +336,7 @@ by simpa only [union_compl_self, restrict_univ]
   using average_union_mem_open_segment ae_disjoint_compl_right hs.compl hs₀ hsc₀
     (measure_ne_top _ _) (measure_ne_top _ _) hfi.integrable_on hfi.integrable_on
 
-@[simp] lemma average_const [is_finite_measure μ] [h : μ.ae.ne_bot] (c : E) :
+@[simp] lemma average_const (μ : measure α) [is_finite_measure μ] [h : μ.ae.ne_bot] (c : E) :
   ⨍ x, c ∂μ = c :=
 by simp only [average_eq, integral_const, measure.restrict_apply, measurable_set.univ, one_smul,
   univ_inter, smul_smul, ← ennreal.to_real_inv, ← ennreal.to_real_mul, ennreal.inv_mul_cancel,
@@ -245,23 +401,23 @@ lemma of_real_set_average {f : α → ℝ} (hf : integrable_on f s μ)
   ennreal.of_real (⨍ x in s, f x ∂μ) = (∫⁻ x in s, ennreal.of_real (f x) ∂μ) / μ s :=
 by simpa using of_real_average hf hf₀
 
-lemma average_to_real {f : α → ℝ≥0∞} (hf : ae_measurable f μ) (hf' : ∀ᵐ x ∂μ, f x ≠ ∞) :
-  ⨍ x, (f x).to_real ∂μ = (∫⁻ x, f x ∂μ / μ univ).to_real :=
+lemma to_real_laverage {f : α → ℝ≥0∞} (hf : ae_measurable f μ) (hf' : ∀ᵐ x ∂μ, f x ≠ ∞) :
+  (⨍⁻ x, f x ∂μ).to_real = ⨍ x, (f x).to_real ∂μ :=
 begin
   obtain rfl | hμ := eq_or_ne μ 0,
   { simp },
-  { rw [average_eq, smul_eq_mul, to_real_div,
-      ←integral_to_real hf (hf'.mono $ λ _, lt_top_iff_ne_top.2), div_eq_inv_mul] }
+  { rw [average_eq, laverage_eq, smul_eq_mul, to_real_div, div_eq_inv_mul,
+      ←integral_to_real hf (hf'.mono $ λ _, lt_top_iff_ne_top.2)] }
 end
 
-lemma set_average_to_real {f : α → ℝ≥0∞} (hf : ae_measurable f (μ.restrict s))
+lemma to_real_set_laverage {f : α → ℝ≥0∞} (hf : ae_measurable f (μ.restrict s))
   (hf' : ∀ᵐ x ∂(μ.restrict s), f x ≠ ∞) :
-  ⨍ x in s, (f x).to_real ∂μ = (∫⁻ x in s, f x ∂μ / μ s).to_real :=
-by simpa using average_to_real hf hf'
+  (∫⁻ x in s, f x ∂μ / μ s).to_real = ⨍ x in s, (f x).to_real ∂μ :=
+by simpa [laverage_eq] using to_real_laverage hf hf'
 
 /-! ### First moment method -/
 
-section first_moment
+section first_moment_real
 variables {N : set α} {f : α → ℝ}
 
 /-- **First moment method**. An integrable function is smaller than its mean on a set of positive
@@ -384,5 +540,158 @@ by simpa only [average_eq_integral]
   using exists_not_mem_null_average_le (is_probability_measure.ne_zero μ) hf hN
 
 end probability_measure
-end first_moment
+end first_moment_real
+
+section first_moment_ennreal
+variables {N : set α} {f : α → ℝ≥0∞}
+
+/-- **First moment method**. A measurable function is smaller than its mean on a set of positive
+measure. -/
+lemma measure_le_set_laverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞)
+  (hf : ae_measurable f (μ.restrict s)) : 0 < μ {x ∈ s | f x ≤ ⨍⁻ a in s, f a ∂μ} :=
+begin
+  obtain h | h := eq_or_ne (∫⁻ a in s, f a ∂μ) ∞,
+  { simpa [mul_top, hμ₁, laverage, h, top_div_of_ne_top hμ₁, pos_iff_ne_zero] using hμ },
+  have := measure_le_set_average_pos hμ hμ₁ (integrable_to_real_of_lintegral_ne_top hf h),
+  rw [←set_of_inter_eq_sep, ←measure.restrict_apply₀
+    (hf.ae_strongly_measurable.null_measurable_set_le ae_strongly_measurable_const)],
+  rw [←set_of_inter_eq_sep, ←measure.restrict_apply₀
+    (hf.ennreal_to_real.ae_strongly_measurable.null_measurable_set_le ae_strongly_measurable_const),
+    ←measure_diff_null (measure_eq_top_of_lintegral_ne_top hf h)] at this,
+  refine this.trans_le (measure_mono _),
+  rintro x ⟨hfx, hx⟩,
+  dsimp at hfx,
+  rwa [←to_real_laverage hf, to_real_le_to_real hx (set_laverage_lt_top h).ne] at hfx,
+  { simp_rw [ae_iff, not_ne_iff],
+    exact measure_eq_top_of_lintegral_ne_top hf h }
+end
+
+/-- **First moment method**. A measurable function is greater than its mean on a set of positive
+measure. -/
+lemma measure_set_laverage_le_pos (hμ : μ s ≠ 0) (hs : null_measurable_set s μ)
+  (hint : ∫⁻ a in s, f a ∂μ ≠ ∞) : 0 < μ {x ∈ s | ⨍⁻ a in s, f a ∂μ ≤ f x} :=
+begin
+  obtain hμ₁ | hμ₁ := eq_or_ne (μ s) ∞,
+  { simp [set_laverage_eq, hμ₁] },
+  obtain ⟨g, hg, hgf, hfg⟩ := exists_measurable_le_lintegral_eq (μ.restrict s) f,
+  have hfg' : ⨍⁻ a in s, f a ∂μ = ⨍⁻ a in s, g a ∂μ,
+  { simp_rw [laverage_eq, hfg] },
+  rw hfg at hint,
+  have := measure_set_average_le_pos hμ hμ₁
+    (integrable_to_real_of_lintegral_ne_top hg.ae_measurable hint),
+  simp_rw [←set_of_inter_eq_sep, ←measure.restrict_apply₀' hs, hfg'],
+  rw [←set_of_inter_eq_sep, ←measure.restrict_apply₀' hs,
+    ←measure_diff_null (measure_eq_top_of_lintegral_ne_top hg.ae_measurable hint)] at this,
+  refine this.trans_le (measure_mono _),
+  rintro x ⟨hfx, hx⟩,
+  dsimp at hfx,
+  rw [←to_real_laverage hg.ae_measurable, to_real_le_to_real (set_laverage_lt_top hint).ne hx]
+    at hfx,
+  exact hfx.trans (hgf _),
+  { simp_rw [ae_iff, not_ne_iff],
+    exact measure_eq_top_of_lintegral_ne_top hg.ae_measurable hint }
+end
+
+/-- **First moment method**. The minimum of a measurable function is smaller than its mean. -/
+lemma exists_le_set_laverage (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : ae_measurable f (μ.restrict s)) :
+  ∃ x ∈ s, f x ≤ ⨍⁻ a in s, f a ∂μ :=
+let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_le_set_laverage_pos hμ hμ₁ hf).ne'
+  in ⟨x, hx, h⟩
+
+/-- **First moment method**. The maximum of a measurable function is greater than its mean. -/
+lemma exists_set_laverage_le (hμ : μ s ≠ 0) (hs : null_measurable_set s μ)
+  (hint : ∫⁻ a in s, f a ∂μ ≠ ∞) : ∃ x ∈ s, ⨍⁻ a in s, f a ∂μ ≤ f x :=
+let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_set_laverage_le_pos hμ hs hint).ne'
+  in ⟨x, hx, h⟩
+
+/-- **First moment method**. A measurable function is greater than its mean on a set of positive
+measure. -/
+lemma measure_laverage_le_pos (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞) :
+  0 < μ {x | ⨍⁻ a, f a ∂μ ≤ f x} :=
+by simpa [hint] using @measure_set_laverage_le_pos _ _ _ _ f (measure_univ_ne_zero.2 hμ)
+  null_measurable_set_univ
+
+/-- **First moment method**. The maximum of a measurable function is greater than its mean. -/
+lemma exists_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞) : ∃ x, ⨍⁻ a, f a ∂μ ≤ f x :=
+let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_laverage_le_pos hμ hint).ne' in ⟨x, hx⟩
+
+/-- **First moment method**. The maximum of a measurable function is greater than its mean, while
+avoiding a null set. -/
+lemma exists_not_mem_null_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ (a : α), f a ∂μ ≠ ∞) (hN : μ N = 0) :
+  ∃ x ∉ N, ⨍⁻ a, f a ∂μ ≤ f x :=
+begin
+  have := measure_laverage_le_pos hμ hint,
+  rw ←measure_diff_null hN at this,
+  obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne',
+  exact ⟨x, hxN, hx⟩,
+end
+
+section finite_measure
+variables [is_finite_measure μ]
+
+/-- **First moment method**. A measurable function is smaller than its mean on a set of positive
+measure. -/
+lemma measure_le_laverage_pos (hμ : μ ≠ 0) (hf : ae_measurable f μ) :
+  0 < μ {x | f x ≤ ⨍⁻ a, f a ∂μ} :=
+by simpa
+  using measure_le_set_laverage_pos (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _) hf.restrict
+
+/-- **First moment method**. The minimum of a measurable function is smaller than its mean. -/
+lemma exists_le_laverage (hμ : μ ≠ 0) (hf : ae_measurable f μ) : ∃ x, f x ≤ ⨍⁻ a, f a ∂μ :=
+let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_le_laverage_pos hμ hf).ne' in ⟨x, hx⟩
+
+/-- **First moment method**. The minimum of a measurable function is smaller than its mean, while
+avoiding a null set. -/
+lemma exists_not_mem_null_le_laverage (hμ : μ ≠ 0) (hf : ae_measurable f μ) (hN : μ N = 0) :
+  ∃ x ∉ N, f x ≤ ⨍⁻ a, f a ∂μ :=
+begin
+  have := measure_le_laverage_pos hμ hf,
+  rw ←measure_diff_null hN at this,
+  obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne',
+  exact ⟨x, hxN, hx⟩,
+end
+
+end finite_measure
+
+section probability_measure
+variables [is_probability_measure μ]
+
+/-- **First moment method**. A measurable function is smaller than its integral on a set of
+positive measure. -/
+lemma measure_le_lintegral_pos (hf : ae_measurable f μ) : 0 < μ {x | f x ≤ ∫⁻ a, f a ∂μ} :=
+by simpa only [laverage_eq_lintegral]
+  using measure_le_laverage_pos (is_probability_measure.ne_zero μ) hf
+
+/-- **First moment method**. A measurable function is greater than its integral on a set of
+positive measure. -/
+lemma measure_lintegral_le_pos (hint : ∫⁻ a, f a ∂μ ≠ ∞) : 0 < μ {x | ∫⁻ a, f a ∂μ ≤ f x} :=
+by simpa only [laverage_eq_lintegral]
+  using measure_laverage_le_pos (is_probability_measure.ne_zero μ) hint
+
+/-- **First moment method**. The minimum of a measurable function is smaller than its integral. -/
+lemma exists_le_lintegral (hf : ae_measurable f μ) : ∃ x, f x ≤ ∫⁻ a, f a ∂μ :=
+by simpa only [laverage_eq_lintegral]
+  using exists_le_laverage (is_probability_measure.ne_zero μ) hf
+
+/-- **First moment method**. The maximum of a measurable function is greater than its integral. -/
+lemma exists_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) : ∃ x, ∫⁻ a, f a ∂μ ≤ f x :=
+by simpa only [laverage_eq_lintegral]
+  using exists_laverage_le (is_probability_measure.ne_zero μ) hint
+
+/-- **First moment method**. The minimum of a measurable function is smaller than its integral,
+while avoiding a null set. -/
+lemma exists_not_mem_null_le_lintegral (hf : ae_measurable f μ) (hN : μ N = 0) :
+  ∃ x ∉ N, f x ≤ ∫⁻ a, f a ∂μ :=
+by simpa only [laverage_eq_lintegral]
+  using exists_not_mem_null_le_laverage (is_probability_measure.ne_zero μ) hf hN
+
+/-- **First moment method**. The maximum of a measurable function is greater than its integral,
+while avoiding a null set. -/
+lemma exists_not_mem_null_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞)
+  (hN : μ N = 0) : ∃ x ∉ N, ∫⁻ a, f a ∂μ ≤ f x :=
+by simpa only [laverage_eq_lintegral]
+  using exists_not_mem_null_laverage_le (is_probability_measure.ne_zero μ) hint hN
+
+end probability_measure
+end first_moment_ennreal
 end measure_theory

feat(measure_theory/integral/set_integral): First moment method (#18731)

Integrable functions are smaller/larger than their mean on a set of positive measure. We prove it for the Bochner and Lebesgue integrals.

Diff

@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Yury G. Kudryashov
+Authors: Yury G. Kudryashov, Yaël Dillies
 -/
 import measure_theory.integral.set_integral
 
@@ -19,24 +19,32 @@ measure, then the average of any function is equal to its integral.
 For the average on a set, we use `⨍ x in s, f x ∂μ` (notation for `⨍ x, f x ∂(μ.restrict s)`). For
 average w.r.t. the volume, one can omit `∂volume`.
 
+We prove several version of the first moment method: An integrable function is below/above its
+average on a set of positive measure.
+
 ## Implementation notes
 
 The average is defined as an integral over `(μ univ)⁻¹ • μ` so that all theorems about Bochner
 integrals work for the average without modifications. For theorems that require integrability of a
 function, we provide a convenience lemma `measure_theory.integrable.to_average`.
 
+## TODO
+
+Provide the first moment method for the Lebesgue integral as well. A draft is available on branch
+`first_moment_lintegral`.
+
 ## Tags
 
 integral, center mass, average value
 -/
 
-open measure_theory measure_theory.measure metric set filter topological_space function
+open ennreal measure_theory measure_theory.measure metric set filter topological_space function
 open_locale topology big_operators ennreal convex
 
 variables {α E F : Type*} {m0 : measurable_space α}
   [normed_add_comm_group E] [normed_space ℝ E] [complete_space E]
   [normed_add_comm_group F] [normed_space ℝ F] [complete_space F]
-  {μ : measure α} {s : set E}
+  {μ : measure α} {s t : set α}
 
 /-!
 ### Average value of a function w.r.t. a measure
@@ -49,8 +57,8 @@ integral.
 -/
 
 namespace measure_theory
-
-variable (μ)
+section normed_add_comm_group
+variables (μ) {f g : α → E}
 include m0
 
 /-- Average value of a function `f` w.r.t. a measure `μ`, notation: `⨍ x, f x ∂μ`. It is defined as
@@ -106,6 +114,9 @@ variable {μ}
 lemma average_congr {f g : α → E} (h : f =ᵐ[μ] g) : ⨍ x, f x ∂μ = ⨍ x, g x ∂μ :=
 by simp only [average_eq, integral_congr_ae h]
 
+lemma set_average_congr_set_ae (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ :=
+by simp only [set_average_eq, set_integral_congr_set_ae h, measure_congr h]
+
 lemma average_add_measure [is_finite_measure μ] {ν : measure α} [is_finite_measure ν] {f : α → E}
   (hμ : integrable f μ) (hν : integrable f ν) :
   ⨍ x, f x ∂(μ + ν) =
@@ -182,4 +193,196 @@ by simp only [set_average_eq, integral_const, measure.restrict_apply, measurable
   univ_inter, smul_smul, ← ennreal.to_real_inv, ← ennreal.to_real_mul,
   ennreal.inv_mul_cancel hs₀ hs, ennreal.one_to_real, one_smul]
 
+@[simp] lemma integral_average (μ : measure α) [is_finite_measure μ] (f : α → E) :
+  ∫ x, ⨍ a, f a ∂μ ∂μ = ∫ x, f x ∂μ :=
+begin
+  unfreezingI { obtain rfl | hμ := eq_or_ne μ 0 },
+  { simp only [integral_zero_measure] },
+  { rw [integral_const, average_eq,
+      smul_inv_smul₀ (to_real_ne_zero.2 ⟨measure_univ_ne_zero.2 hμ, measure_ne_top _ _⟩)] }
+end
+
+lemma set_integral_set_average (μ : measure α) [is_finite_measure μ] (f : α → E) (s : set α) :
+  ∫ x in s, ⨍ a in s, f a ∂μ ∂μ = ∫ x in s, f x ∂μ :=
+integral_average _ _
+
+lemma integral_sub_average (μ : measure α) [is_finite_measure μ] (f : α → E) :
+  ∫ x, f x - ⨍ a, f a ∂μ ∂μ = 0 :=
+begin
+  by_cases hf : integrable f μ,
+  { rw [integral_sub hf (integrable_const _), integral_average, sub_self] },
+  refine integral_undef (λ h, hf _),
+  convert h.add (integrable_const _),
+  exact (sub_add_cancel _ _).symm,
+end
+
+lemma set_integral_sub_set_average (hs : μ s ≠ ∞) (f : α → E) :
+  ∫ x in s, f x - ⨍ a in s, f a ∂μ ∂μ = 0 :=
+by haveI haveI : fact (μ s < ∞) := ⟨lt_top_iff_ne_top.2 hs⟩; exact integral_sub_average _ _
+
+lemma integral_average_sub [is_finite_measure μ] (hf : integrable f μ) :
+  ∫ x, ⨍ a, f a ∂μ - f x ∂μ = 0 :=
+by rw [integral_sub (integrable_const _) hf, integral_average, sub_self]
+
+lemma set_integral_set_average_sub (hs : μ s ≠ ∞) (hf : integrable_on f s μ) :
+  ∫ x in s, ⨍ a in s, f a ∂μ - f x ∂μ = 0 :=
+by haveI haveI : fact (μ s < ∞) := ⟨lt_top_iff_ne_top.2 hs⟩; exact integral_average_sub hf
+
+end normed_add_comm_group
+
+lemma of_real_average {f : α → ℝ} (hf : integrable f μ) (hf₀ : 0 ≤ᵐ[μ] f) :
+  ennreal.of_real (⨍ x, f x ∂μ) = (∫⁻ x, ennreal.of_real (f x) ∂μ) / μ univ :=
+begin
+  obtain rfl | hμ := eq_or_ne μ 0,
+  { simp },
+  { rw [average_eq, smul_eq_mul, ←to_real_inv, of_real_mul (to_real_nonneg),
+      of_real_to_real (inv_ne_top.2 $ measure_univ_ne_zero.2 hμ),
+      of_real_integral_eq_lintegral_of_real hf hf₀, ennreal.div_eq_inv_mul] }
+end
+
+lemma of_real_set_average {f : α → ℝ} (hf : integrable_on f s μ)
+  (hf₀ : 0 ≤ᵐ[μ.restrict s] f) :
+  ennreal.of_real (⨍ x in s, f x ∂μ) = (∫⁻ x in s, ennreal.of_real (f x) ∂μ) / μ s :=
+by simpa using of_real_average hf hf₀
+
+lemma average_to_real {f : α → ℝ≥0∞} (hf : ae_measurable f μ) (hf' : ∀ᵐ x ∂μ, f x ≠ ∞) :
+  ⨍ x, (f x).to_real ∂μ = (∫⁻ x, f x ∂μ / μ univ).to_real :=
+begin
+  obtain rfl | hμ := eq_or_ne μ 0,
+  { simp },
+  { rw [average_eq, smul_eq_mul, to_real_div,
+      ←integral_to_real hf (hf'.mono $ λ _, lt_top_iff_ne_top.2), div_eq_inv_mul] }
+end
+
+lemma set_average_to_real {f : α → ℝ≥0∞} (hf : ae_measurable f (μ.restrict s))
+  (hf' : ∀ᵐ x ∂(μ.restrict s), f x ≠ ∞) :
+  ⨍ x in s, (f x).to_real ∂μ = (∫⁻ x in s, f x ∂μ / μ s).to_real :=
+by simpa using average_to_real hf hf'
+
+/-! ### First moment method -/
+
+section first_moment
+variables {N : set α} {f : α → ℝ}
+
+/-- **First moment method**. An integrable function is smaller than its mean on a set of positive
+measure. -/
+lemma measure_le_set_average_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : integrable_on f s μ) :
+  0 < μ {x ∈ s | f x ≤ ⨍ a in s, f a ∂μ} :=
+begin
+  refine pos_iff_ne_zero.2 (λ H, _),
+  replace H : (μ.restrict s) {x | f x ≤ ⨍ a in s, f a ∂μ} = 0,
+  { rwa [restrict_apply₀, inter_comm],
+    exact ae_strongly_measurable.null_measurable_set_le hf.1 ae_strongly_measurable_const },
+  haveI : is_finite_measure (μ.restrict s) :=
+    ⟨by simpa only [measure.restrict_apply, measurable_set.univ, univ_inter] using hμ₁.lt_top⟩,
+  refine (integral_sub_average (μ.restrict s) f).not_gt _,
+  refine (set_integral_pos_iff_support_of_nonneg_ae _ _).2 _,
+  { refine eq_bot_mono (measure_mono (λ x hx, _)) H,
+    simp only [pi.zero_apply, sub_nonneg, mem_compl_iff, mem_set_of_eq, not_le] at hx,
+    exact hx.le },
+  { exact hf.sub (integrable_on_const.2 $ or.inr $ lt_top_iff_ne_top.2 hμ₁) },
+  { rwa [pos_iff_ne_zero, inter_comm, ←diff_compl, ←diff_inter_self_eq_diff, measure_diff_null],
+    refine eq_bot_mono (measure_mono _) (measure_inter_eq_zero_of_restrict H),
+    exact inter_subset_inter_left _ (λ a ha, (sub_eq_zero.1 $ of_not_not ha).le) }
+end
+
+/-- **First moment method**. An integrable function is greater than its mean on a set of positive
+measure. -/
+lemma measure_set_average_le_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : integrable_on f s μ) :
+  0 < μ {x ∈ s | ⨍ a in s, f a ∂μ ≤ f x} :=
+by simpa [integral_neg, neg_div] using measure_le_set_average_pos hμ hμ₁ hf.neg
+
+/-- **First moment method**. The minimum of an integrable function is smaller than its mean. -/
+lemma exists_le_set_average (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : integrable_on f s μ) :
+  ∃ x ∈ s, f x ≤ ⨍ a in s, f a ∂μ :=
+let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_le_set_average_pos hμ hμ₁ hf).ne'
+  in ⟨x, hx, h⟩
+
+/-- **First moment method**. The maximum of an integrable function is greater than its mean. -/
+lemma exists_set_average_le (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : integrable_on f s μ) :
+  ∃ x ∈ s, ⨍ a in s, f a ∂μ ≤ f x :=
+let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_set_average_le_pos hμ hμ₁ hf).ne'
+  in ⟨x, hx, h⟩
+
+section finite_measure
+variables [is_finite_measure μ]
+
+/-- **First moment method**. An integrable function is smaller than its mean on a set of positive
+measure. -/
+lemma measure_le_average_pos (hμ : μ ≠ 0) (hf : integrable f μ) : 0 < μ {x | f x ≤ ⨍ a, f a ∂μ} :=
+by simpa using measure_le_set_average_pos (measure.measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)
+  hf.integrable_on
+
+/-- **First moment method**. An integrable function is greater than its mean on a set of positive
+measure. -/
+lemma measure_average_le_pos (hμ : μ ≠ 0) (hf : integrable f μ) : 0 < μ {x | ⨍ a, f a ∂μ ≤ f x} :=
+by simpa using measure_set_average_le_pos (measure.measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)
+  hf.integrable_on
+
+/-- **First moment method**. The minimum of an integrable function is smaller than its mean. -/
+lemma exists_le_average (hμ : μ ≠ 0) (hf : integrable f μ) : ∃ x, f x ≤ ⨍ a, f a ∂μ :=
+let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_le_average_pos hμ hf).ne' in ⟨x, hx⟩
+
+/-- **First moment method**. The maximum of an integrable function is greater than its mean. -/
+lemma exists_average_le (hμ : μ ≠ 0) (hf : integrable f μ) : ∃ x, ⨍ a, f a ∂μ ≤ f x :=
+let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_average_le_pos hμ hf).ne' in ⟨x, hx⟩
+
+/-- **First moment method**. The minimum of an integrable function is smaller than its mean, while
+avoiding a null set. -/
+lemma exists_not_mem_null_le_average (hμ : μ ≠ 0) (hf : integrable f μ) (hN : μ N = 0) :
+  ∃ x ∉ N, f x ≤ ⨍ a, f a ∂μ :=
+begin
+  have := measure_le_average_pos hμ hf,
+  rw ←measure_diff_null hN at this,
+  obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne',
+  exact ⟨x, hxN, hx⟩,
+end
+
+/-- **First moment method**. The maximum of an integrable function is greater than its mean, while
+avoiding a null set. -/
+lemma exists_not_mem_null_average_le (hμ : μ ≠ 0) (hf : integrable f μ) (hN : μ N = 0) :
+  ∃ x ∉ N, ⨍ a, f a ∂μ ≤ f x :=
+by simpa [integral_neg, neg_div] using exists_not_mem_null_le_average hμ hf.neg hN
+
+end finite_measure
+
+section probability_measure
+variables [is_probability_measure μ]
+
+/-- **First moment method**. An integrable function is smaller than its integral on a set of
+positive measure. -/
+lemma measure_le_integral_pos (hf : integrable f μ) : 0 < μ {x | f x ≤ ∫ a, f a ∂μ} :=
+by simpa only [average_eq_integral]
+  using measure_le_average_pos (is_probability_measure.ne_zero μ) hf
+
+/-- **First moment method**. An integrable function is greater than its integral on a set of
+positive measure. -/
+lemma measure_integral_le_pos (hf : integrable f μ) : 0 < μ {x | ∫ a, f a ∂μ ≤ f x} :=
+by simpa only [average_eq_integral]
+  using measure_average_le_pos (is_probability_measure.ne_zero μ) hf
+
+/-- **First moment method**. The minimum of an integrable function is smaller than its integral. -/
+lemma exists_le_integral (hf : integrable f μ) : ∃ x, f x ≤ ∫ a, f a ∂μ :=
+by simpa only [average_eq_integral] using exists_le_average (is_probability_measure.ne_zero μ) hf
+
+/-- **First moment method**. The maximum of an integrable function is greater than its integral. -/
+lemma exists_integral_le (hf : integrable f μ) : ∃ x, ∫ a, f a ∂μ ≤ f x :=
+by simpa only [average_eq_integral] using exists_average_le (is_probability_measure.ne_zero μ) hf
+
+/-- **First moment method**. The minimum of an integrable function is smaller than its integral,
+while avoiding a null set. -/
+lemma exists_not_mem_null_le_integral (hf : integrable f μ) (hN : μ N = 0) :
+  ∃ x ∉ N, f x ≤ ∫ a, f a ∂μ :=
+by simpa only [average_eq_integral]
+  using exists_not_mem_null_le_average (is_probability_measure.ne_zero μ) hf hN
+
+/-- **First moment method**. The maximum of an integrable function is greater than its integral,
+while avoiding a null set. -/
+lemma exists_not_mem_null_integral_le (hf : integrable f μ) (hN : μ N = 0) :
+  ∃ x ∉ N, ∫ a, f a ∂μ ≤ f x :=
+by simpa only [average_eq_integral]
+  using exists_not_mem_null_average_le (is_probability_measure.ne_zero μ) hf hN
+
+end probability_measure
+end first_moment
 end measure_theory

(first ported)

Changes in mathlib3port

mathlib3

mathlib3port

bump to nightly-2024-04-06-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

e34029c9

Diff

@@ -705,7 +705,7 @@ theorem exists_average_le (hμ : μ ≠ 0) (hf : Integrable f μ) : ∃ x, ⨍ a
 #align measure_theory.exists_average_le MeasureTheory.exists_average_le
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_le_average /-
 /-- **First moment method**. The minimum of an integrable function is smaller than its mean, while
 avoiding a null set. -/
@@ -719,7 +719,7 @@ theorem exists_not_mem_null_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) (
 #align measure_theory.exists_not_mem_null_le_average MeasureTheory.exists_not_mem_null_le_average
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_average_le /-
 /-- **First moment method**. The maximum of an integrable function is greater than its mean, while
 avoiding a null set. -/
@@ -767,7 +767,7 @@ theorem exists_integral_le (hf : Integrable f μ) : ∃ x, ∫ a, f a ∂μ ≤
 #align measure_theory.exists_integral_le MeasureTheory.exists_integral_le
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_le_integral /-
 /-- **First moment method**. The minimum of an integrable function is smaller than its integral,
 while avoiding a null set. -/
@@ -778,7 +778,7 @@ theorem exists_not_mem_null_le_integral (hf : Integrable f μ) (hN : μ N = 0) :
 #align measure_theory.exists_not_mem_null_le_integral MeasureTheory.exists_not_mem_null_le_integral
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_integral_le /-
 /-- **First moment method**. The maximum of an integrable function is greater than its integral,
 while avoiding a null set. -/
@@ -886,7 +886,7 @@ theorem exists_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞)
 #align measure_theory.exists_laverage_le MeasureTheory.exists_laverage_le
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_laverage_le /-
 /-- **First moment method**. The maximum of a measurable function is greater than its mean, while
 avoiding a null set. -/
@@ -922,7 +922,7 @@ theorem exists_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) : ∃ x, f
 #align measure_theory.exists_le_laverage MeasureTheory.exists_le_laverage
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_le_laverage /-
 /-- **First moment method**. The minimum of a measurable function is smaller than its mean, while
 avoiding a null set. -/
@@ -975,7 +975,7 @@ theorem exists_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) : ∃ x, ∫⁻
 #align measure_theory.exists_lintegral_le MeasureTheory.exists_lintegral_le
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_le_lintegral /-
 /-- **First moment method**. The minimum of a measurable function is smaller than its integral,
 while avoiding a null set. -/
@@ -986,7 +986,7 @@ theorem exists_not_mem_null_le_lintegral (hf : AEMeasurable f μ) (hN : μ N = 0
 #align measure_theory.exists_not_mem_null_le_lintegral MeasureTheory.exists_not_mem_null_le_lintegral
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_lintegral_le /-
 /-- **First moment method**. The maximum of a measurable function is greater than its integral,
 while avoiding a null set. -/

bump to nightly-2024-03-17-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

22f01ce4

Diff

@@ -180,10 +180,10 @@ theorem laverage_add_measure :
       μ univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂μ + ν univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂ν :=
   by
   by_cases hμ : is_finite_measure μ; swap
-  · rw [not_is_finite_measure_iff] at hμ 
+  · rw [not_is_finite_measure_iff] at hμ
     simp [laverage_eq, hμ]
   by_cases hν : is_finite_measure ν; swap
-  · rw [not_is_finite_measure_iff] at hν 
+  · rw [not_is_finite_measure_iff] at hν
     simp [laverage_eq, hν]
   haveI := hμ; haveI := hν
   simp only [← ENNReal.mul_div_right_comm, measure_mul_laverage, ← ENNReal.add_div, ←
@@ -225,7 +225,7 @@ theorem laverage_union_mem_segment (hd : AEDisjoint μ s t) (ht : NullMeasurable
     ⨍⁻ x in s ∪ t, f x ∂μ ∈ [⨍⁻ x in s, f x ∂μ -[ℝ≥0∞] ⨍⁻ x in t, f x ∂μ] :=
   by
   by_cases hs₀ : μ s = 0
-  · rw [← ae_eq_empty] at hs₀ 
+  · rw [← ae_eq_empty] at hs₀
     rw [restrict_congr_set (hs₀.union eventually_eq.rfl), empty_union]
     exact right_mem_segment _ _ _
   · refine'
@@ -465,7 +465,7 @@ theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint
     (hft : IntegrableOn f t μ) : ⨍ x in s ∪ t, f x ∂μ ∈ [⨍ x in s, f x ∂μ -[ℝ] ⨍ x in t, f x ∂μ] :=
   by
   by_cases hse : μ s = 0
-  · rw [← ae_eq_empty] at hse 
+  · rw [← ae_eq_empty] at hse
     rw [restrict_congr_set (hse.union eventually_eq.rfl), empty_union]
     exact right_mem_segment _ _ _
   · refine'
@@ -627,7 +627,7 @@ theorem measure_le_setAverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf
   refine' (integral_sub_average (μ.restrict s) f).not_gt _
   refine' (set_integral_pos_iff_support_of_nonneg_ae _ _).2 _
   · refine' eq_bot_mono (measure_mono fun x hx => _) H
-    simp only [Pi.zero_apply, sub_nonneg, mem_compl_iff, mem_set_of_eq, not_le] at hx 
+    simp only [Pi.zero_apply, sub_nonneg, mem_compl_iff, mem_set_of_eq, not_le] at hx
     exact hx.le
   · exact hf.sub (integrable_on_const.2 <| Or.inr <| lt_top_iff_ne_top.2 hμ₁)
   · rwa [pos_iff_ne_zero, inter_comm, ← diff_compl, ← diff_inter_self_eq_diff, measure_diff_null]
@@ -713,7 +713,7 @@ theorem exists_not_mem_null_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) (
     ∃ (x : _) (_ : x ∉ N), f x ≤ ⨍ a, f a ∂μ :=
   by
   have := measure_le_average_pos hμ hf
-  rw [← measure_diff_null hN] at this 
+  rw [← measure_diff_null hN] at this
   obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
   exact ⟨x, hxN, hx⟩
 #align measure_theory.exists_not_mem_null_le_average MeasureTheory.exists_not_mem_null_le_average
@@ -813,11 +813,11 @@ theorem measure_le_setLaverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞)
     measure.restrict_apply₀
       (hf.ennreal_to_real.ae_strongly_measurable.null_measurable_set_le
         ae_strongly_measurable_const),
-    ← measure_diff_null (measure_eq_top_of_lintegral_ne_top hf h)] at this 
+    ← measure_diff_null (measure_eq_top_of_lintegral_ne_top hf h)] at this
   refine' this.trans_le (measure_mono _)
   rintro x ⟨hfx, hx⟩
-  dsimp at hfx 
-  rwa [← to_real_laverage hf, to_real_le_to_real hx (set_laverage_lt_top h).Ne] at hfx 
+  dsimp at hfx
+  rwa [← to_real_laverage hf, to_real_le_to_real hx (set_laverage_lt_top h).Ne] at hfx
   · simp_rw [ae_iff, not_ne_iff]
     exact measure_eq_top_of_lintegral_ne_top hf h
 #align measure_theory.measure_le_set_laverage_pos MeasureTheory.measure_le_setLaverage_pos
@@ -833,17 +833,17 @@ theorem measure_setLaverage_le_pos (hμ : μ s ≠ 0) (hs : NullMeasurableSet s
   · simp [set_laverage_eq, hμ₁]
   obtain ⟨g, hg, hgf, hfg⟩ := exists_measurable_le_lintegral_eq (μ.restrict s) f
   have hfg' : ⨍⁻ a in s, f a ∂μ = ⨍⁻ a in s, g a ∂μ := by simp_rw [laverage_eq, hfg]
-  rw [hfg] at hint 
+  rw [hfg] at hint
   have :=
     measure_set_average_le_pos hμ hμ₁ (integrable_to_real_of_lintegral_ne_top hg.ae_measurable hint)
   simp_rw [← set_of_inter_eq_sep, ← measure.restrict_apply₀' hs, hfg']
   rw [← set_of_inter_eq_sep, ← measure.restrict_apply₀' hs, ←
-    measure_diff_null (measure_eq_top_of_lintegral_ne_top hg.ae_measurable hint)] at this 
+    measure_diff_null (measure_eq_top_of_lintegral_ne_top hg.ae_measurable hint)] at this
   refine' this.trans_le (measure_mono _)
   rintro x ⟨hfx, hx⟩
-  dsimp at hfx 
+  dsimp at hfx
   rw [← to_real_laverage hg.ae_measurable, to_real_le_to_real (set_laverage_lt_top hint).Ne hx] at
-    hfx 
+    hfx
   exact hfx.trans (hgf _)
   · simp_rw [ae_iff, not_ne_iff]
     exact measure_eq_top_of_lintegral_ne_top hg.ae_measurable hint
@@ -894,7 +894,7 @@ theorem exists_not_mem_null_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a : α,
     ∃ (x : _) (_ : x ∉ N), ⨍⁻ a, f a ∂μ ≤ f x :=
   by
   have := measure_laverage_le_pos hμ hint
-  rw [← measure_diff_null hN] at this 
+  rw [← measure_diff_null hN] at this
   obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
   exact ⟨x, hxN, hx⟩
 #align measure_theory.exists_not_mem_null_laverage_le MeasureTheory.exists_not_mem_null_laverage_le
@@ -930,7 +930,7 @@ theorem exists_not_mem_null_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ
     ∃ (x : _) (_ : x ∉ N), f x ≤ ⨍⁻ a, f a ∂μ :=
   by
   have := measure_le_laverage_pos hμ hf
-  rw [← measure_diff_null hN] at this 
+  rw [← measure_diff_null hN] at this
   obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
   exact ⟨x, hxN, hx⟩
 #align measure_theory.exists_not_mem_null_le_laverage MeasureTheory.exists_not_mem_null_le_laverage

bump to nightly-2023-09-24-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

5655435f

Diff

@@ -3,7 +3,7 @@ Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov, Yaël Dillies
 -/
-import Mathbin.MeasureTheory.Integral.SetIntegral
+import MeasureTheory.Integral.SetIntegral
 
 #align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
 
@@ -705,7 +705,7 @@ theorem exists_average_le (hμ : μ ≠ 0) (hf : Integrable f μ) : ∃ x, ⨍ a
 #align measure_theory.exists_average_le MeasureTheory.exists_average_le
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_le_average /-
 /-- **First moment method**. The minimum of an integrable function is smaller than its mean, while
 avoiding a null set. -/
@@ -719,7 +719,7 @@ theorem exists_not_mem_null_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) (
 #align measure_theory.exists_not_mem_null_le_average MeasureTheory.exists_not_mem_null_le_average
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_average_le /-
 /-- **First moment method**. The maximum of an integrable function is greater than its mean, while
 avoiding a null set. -/
@@ -767,7 +767,7 @@ theorem exists_integral_le (hf : Integrable f μ) : ∃ x, ∫ a, f a ∂μ ≤
 #align measure_theory.exists_integral_le MeasureTheory.exists_integral_le
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_le_integral /-
 /-- **First moment method**. The minimum of an integrable function is smaller than its integral,
 while avoiding a null set. -/
@@ -778,7 +778,7 @@ theorem exists_not_mem_null_le_integral (hf : Integrable f μ) (hN : μ N = 0) :
 #align measure_theory.exists_not_mem_null_le_integral MeasureTheory.exists_not_mem_null_le_integral
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_integral_le /-
 /-- **First moment method**. The maximum of an integrable function is greater than its integral,
 while avoiding a null set. -/
@@ -886,7 +886,7 @@ theorem exists_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞)
 #align measure_theory.exists_laverage_le MeasureTheory.exists_laverage_le
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_laverage_le /-
 /-- **First moment method**. The maximum of a measurable function is greater than its mean, while
 avoiding a null set. -/
@@ -922,7 +922,7 @@ theorem exists_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) : ∃ x, f
 #align measure_theory.exists_le_laverage MeasureTheory.exists_le_laverage
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_le_laverage /-
 /-- **First moment method**. The minimum of a measurable function is smaller than its mean, while
 avoiding a null set. -/
@@ -975,7 +975,7 @@ theorem exists_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) : ∃ x, ∫⁻
 #align measure_theory.exists_lintegral_le MeasureTheory.exists_lintegral_le
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_le_lintegral /-
 /-- **First moment method**. The minimum of a measurable function is smaller than its integral,
 while avoiding a null set. -/
@@ -986,7 +986,7 @@ theorem exists_not_mem_null_le_lintegral (hf : AEMeasurable f μ) (hN : μ N = 0
 #align measure_theory.exists_not_mem_null_le_lintegral MeasureTheory.exists_not_mem_null_le_lintegral
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_lintegral_le /-
 /-- **First moment method**. The maximum of a measurable function is greater than its integral,
 while avoiding a null set. -/

bump to nightly-2023-07-20-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

9c56d0dd

Diff

@@ -2,14 +2,11 @@
 Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov, Yaël Dillies
-
-! This file was ported from Lean 3 source module measure_theory.integral.average
-! leanprover-community/mathlib commit c14c8fcde993801fca8946b0d80131a1a81d1520
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.MeasureTheory.Integral.SetIntegral
 
+#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
+
 /-!
 # Integral average of a function
 
@@ -708,7 +705,7 @@ theorem exists_average_le (hμ : μ ≠ 0) (hf : Integrable f μ) : ∃ x, ⨍ a
 #align measure_theory.exists_average_le MeasureTheory.exists_average_le
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_le_average /-
 /-- **First moment method**. The minimum of an integrable function is smaller than its mean, while
 avoiding a null set. -/
@@ -722,7 +719,7 @@ theorem exists_not_mem_null_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) (
 #align measure_theory.exists_not_mem_null_le_average MeasureTheory.exists_not_mem_null_le_average
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_average_le /-
 /-- **First moment method**. The maximum of an integrable function is greater than its mean, while
 avoiding a null set. -/
@@ -770,7 +767,7 @@ theorem exists_integral_le (hf : Integrable f μ) : ∃ x, ∫ a, f a ∂μ ≤
 #align measure_theory.exists_integral_le MeasureTheory.exists_integral_le
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_le_integral /-
 /-- **First moment method**. The minimum of an integrable function is smaller than its integral,
 while avoiding a null set. -/
@@ -781,7 +778,7 @@ theorem exists_not_mem_null_le_integral (hf : Integrable f μ) (hN : μ N = 0) :
 #align measure_theory.exists_not_mem_null_le_integral MeasureTheory.exists_not_mem_null_le_integral
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_integral_le /-
 /-- **First moment method**. The maximum of an integrable function is greater than its integral,
 while avoiding a null set. -/
@@ -889,7 +886,7 @@ theorem exists_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞)
 #align measure_theory.exists_laverage_le MeasureTheory.exists_laverage_le
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_laverage_le /-
 /-- **First moment method**. The maximum of a measurable function is greater than its mean, while
 avoiding a null set. -/
@@ -925,7 +922,7 @@ theorem exists_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) : ∃ x, f
 #align measure_theory.exists_le_laverage MeasureTheory.exists_le_laverage
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_le_laverage /-
 /-- **First moment method**. The minimum of a measurable function is smaller than its mean, while
 avoiding a null set. -/
@@ -978,7 +975,7 @@ theorem exists_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) : ∃ x, ∫⁻
 #align measure_theory.exists_lintegral_le MeasureTheory.exists_lintegral_le
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_le_lintegral /-
 /-- **First moment method**. The minimum of a measurable function is smaller than its integral,
 while avoiding a null set. -/
@@ -989,7 +986,7 @@ theorem exists_not_mem_null_le_lintegral (hf : AEMeasurable f μ) (hN : μ N = 0
 #align measure_theory.exists_not_mem_null_le_lintegral MeasureTheory.exists_not_mem_null_le_lintegral
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » N) -/
 #print MeasureTheory.exists_not_mem_null_lintegral_le /-
 /-- **First moment method**. The maximum of a measurable function is greater than its integral,
 while avoiding a null set. -/

bump to nightly-2023-07-18-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/bf2428c9486c407ca38b5b3fb10b87dad0bc99fa

a4aa5276

Diff

@@ -66,6 +66,7 @@ section ENNReal
 
 variable (μ) {f g : α → ℝ≥0∞}
 
+#print MeasureTheory.laverage /-
 /-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. a measure `μ`, notation: `⨍⁻ x, f x ∂μ`.
 It is defined as `μ univ⁻¹ * ∫⁻ x, f x ∂μ`, so it is equal to zero if `μ` is an infinite measure. If
 `μ` is a probability measure, then the average of any function is equal to its integral.
@@ -75,6 +76,7 @@ average w.r.t. the volume, one can omit `∂volume`. -/
 noncomputable def laverage (f : α → ℝ≥0∞) :=
   ∫⁻ x, f x ∂(μ univ)⁻¹ • μ
 #align measure_theory.laverage MeasureTheory.laverage
+-/
 
 notation3"⨍⁻ "(...)", "r:60:(scoped f => f)" ∂"μ:70 => laverage μ r
 
@@ -84,26 +86,37 @@ notation3"⨍⁻ "(...)" in "s", "r:60:(scoped f => f)" ∂"μ:70 => laverage (M
 
 notation3"⨍⁻ "(...)" in "s", "r:60:(scoped f => laverage Measure.restrict volume s f) => r
 
+#print MeasureTheory.laverage_zero /-
 @[simp]
 theorem laverage_zero : ⨍⁻ x, (0 : ℝ≥0∞) ∂μ = 0 := by rw [laverage, lintegral_zero]
 #align measure_theory.laverage_zero MeasureTheory.laverage_zero
+-/
 
+#print MeasureTheory.laverage_zero_measure /-
 @[simp]
 theorem laverage_zero_measure (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂(0 : Measure α) = 0 := by simp [laverage]
 #align measure_theory.laverage_zero_measure MeasureTheory.laverage_zero_measure
+-/
 
+#print MeasureTheory.laverage_eq' /-
 theorem laverage_eq' (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂(μ univ)⁻¹ • μ :=
   rfl
 #align measure_theory.laverage_eq' MeasureTheory.laverage_eq'
+-/
 
+#print MeasureTheory.laverage_eq /-
 theorem laverage_eq (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = (∫⁻ x, f x ∂μ) / μ univ := by
   rw [laverage_eq', lintegral_smul_measure, ENNReal.div_eq_inv_mul]
 #align measure_theory.laverage_eq MeasureTheory.laverage_eq
+-/
 
+#print MeasureTheory.laverage_eq_lintegral /-
 theorem laverage_eq_lintegral [IsProbabilityMeasure μ] (f : α → ℝ≥0∞) :
     ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ := by rw [laverage, measure_univ, inv_one, one_smul]
 #align measure_theory.laverage_eq_lintegral MeasureTheory.laverage_eq_lintegral
+-/
 
+#print MeasureTheory.measure_mul_laverage /-
 @[simp]
 theorem measure_mul_laverage [IsFiniteMeasure μ] (f : α → ℝ≥0∞) :
     μ univ * ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ :=
@@ -112,31 +125,43 @@ theorem measure_mul_laverage [IsFiniteMeasure μ] (f : α → ℝ≥0∞) :
   · rw [hμ, lintegral_zero_measure, laverage_zero_measure, MulZeroClass.mul_zero]
   · rw [laverage_eq, ENNReal.mul_div_cancel' (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)]
 #align measure_theory.measure_mul_laverage MeasureTheory.measure_mul_laverage
+-/
 
-theorem set_laverage_eq (f : α → ℝ≥0∞) (s : Set α) :
-    ⨍⁻ x in s, f x ∂μ = (∫⁻ x in s, f x ∂μ) / μ s := by rw [laverage_eq, restrict_apply_univ]
-#align measure_theory.set_laverage_eq MeasureTheory.set_laverage_eq
+#print MeasureTheory.setLaverage_eq /-
+theorem setLaverage_eq (f : α → ℝ≥0∞) (s : Set α) : ⨍⁻ x in s, f x ∂μ = (∫⁻ x in s, f x ∂μ) / μ s :=
+  by rw [laverage_eq, restrict_apply_univ]
+#align measure_theory.set_laverage_eq MeasureTheory.setLaverage_eq
+-/
 
-theorem set_laverage_eq' (f : α → ℝ≥0∞) (s : Set α) :
+#print MeasureTheory.setLaverage_eq' /-
+theorem setLaverage_eq' (f : α → ℝ≥0∞) (s : Set α) :
     ⨍⁻ x in s, f x ∂μ = ∫⁻ x, f x ∂(μ s)⁻¹ • μ.restrict s := by
   simp only [laverage_eq', restrict_apply_univ]
-#align measure_theory.set_laverage_eq' MeasureTheory.set_laverage_eq'
+#align measure_theory.set_laverage_eq' MeasureTheory.setLaverage_eq'
+-/
 
 variable {μ}
 
+#print MeasureTheory.laverage_congr /-
 theorem laverage_congr {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : ⨍⁻ x, f x ∂μ = ⨍⁻ x, g x ∂μ := by
   simp only [laverage_eq, lintegral_congr_ae h]
 #align measure_theory.laverage_congr MeasureTheory.laverage_congr
+-/
 
-theorem set_laverage_congr (h : s =ᵐ[μ] t) : ⨍⁻ x in s, f x ∂μ = ⨍⁻ x in t, f x ∂μ := by
+#print MeasureTheory.setLaverage_congr /-
+theorem setLaverage_congr (h : s =ᵐ[μ] t) : ⨍⁻ x in s, f x ∂μ = ⨍⁻ x in t, f x ∂μ := by
   simp only [set_laverage_eq, set_lintegral_congr h, measure_congr h]
-#align measure_theory.set_laverage_congr MeasureTheory.set_laverage_congr
+#align measure_theory.set_laverage_congr MeasureTheory.setLaverage_congr
+-/
 
-theorem set_laverage_congr_fun (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
+#print MeasureTheory.setLaverage_congr_fun /-
+theorem setLaverage_congr_fun (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
     ⨍⁻ x in s, f x ∂μ = ⨍⁻ x in s, g x ∂μ := by
   simp only [laverage_eq, set_lintegral_congr_fun hs h]
-#align measure_theory.set_laverage_congr_fun MeasureTheory.set_laverage_congr_fun
+#align measure_theory.set_laverage_congr_fun MeasureTheory.setLaverage_congr_fun
+-/
 
+#print MeasureTheory.laverage_lt_top /-
 theorem laverage_lt_top (hf : ∫⁻ x, f x ∂μ ≠ ∞) : ⨍⁻ x, f x ∂μ < ∞ :=
   by
   obtain rfl | hμ := eq_or_ne μ 0
@@ -144,11 +169,15 @@ theorem laverage_lt_top (hf : ∫⁻ x, f x ∂μ ≠ ∞) : ⨍⁻ x, f x ∂μ
   · rw [laverage_eq]
     exact div_lt_top hf (measure_univ_ne_zero.2 hμ)
 #align measure_theory.laverage_lt_top MeasureTheory.laverage_lt_top
+-/
 
-theorem set_laverage_lt_top : ∫⁻ x in s, f x ∂μ ≠ ∞ → ⨍⁻ x in s, f x ∂μ < ∞ :=
+#print MeasureTheory.setLaverage_lt_top /-
+theorem setLaverage_lt_top : ∫⁻ x in s, f x ∂μ ≠ ∞ → ⨍⁻ x in s, f x ∂μ < ∞ :=
   laverage_lt_top
-#align measure_theory.set_laverage_lt_top MeasureTheory.set_laverage_lt_top
+#align measure_theory.set_laverage_lt_top MeasureTheory.setLaverage_lt_top
+-/
 
+#print MeasureTheory.laverage_add_measure /-
 theorem laverage_add_measure :
     ⨍⁻ x, f x ∂(μ + ν) =
       μ univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂μ + ν univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂ν :=
@@ -163,18 +192,24 @@ theorem laverage_add_measure :
   simp only [← ENNReal.mul_div_right_comm, measure_mul_laverage, ← ENNReal.add_div, ←
     lintegral_add_measure, ← measure.add_apply, ← laverage_eq]
 #align measure_theory.laverage_add_measure MeasureTheory.laverage_add_measure
+-/
 
-theorem measure_mul_set_laverage (f : α → ℝ≥0∞) (h : μ s ≠ ∞) :
+#print MeasureTheory.measure_mul_setLaverage /-
+theorem measure_mul_setLaverage (f : α → ℝ≥0∞) (h : μ s ≠ ∞) :
     μ s * ⨍⁻ x in s, f x ∂μ = ∫⁻ x in s, f x ∂μ := by haveI := Fact.mk h.lt_top;
   rw [← measure_mul_laverage, restrict_apply_univ]
-#align measure_theory.measure_mul_set_laverage MeasureTheory.measure_mul_set_laverage
+#align measure_theory.measure_mul_set_laverage MeasureTheory.measure_mul_setLaverage
+-/
 
+#print MeasureTheory.laverage_union /-
 theorem laverage_union (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ) :
     ⨍⁻ x in s ∪ t, f x ∂μ =
       μ s / (μ s + μ t) * ⨍⁻ x in s, f x ∂μ + μ t / (μ s + μ t) * ⨍⁻ x in t, f x ∂μ :=
   by rw [restrict_union₀ hd ht, laverage_add_measure, restrict_apply_univ, restrict_apply_univ]
 #align measure_theory.laverage_union MeasureTheory.laverage_union
+-/
 
+#print MeasureTheory.laverage_union_mem_openSegment /-
 theorem laverage_union_mem_openSegment (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
     (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) :
     ⨍⁻ x in s ∪ t, f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ x in s, f x ∂μ) (⨍⁻ x in t, f x ∂μ) :=
@@ -185,7 +220,9 @@ theorem laverage_union_mem_openSegment (hd : AEDisjoint μ s t) (ht : NullMeasur
   rw [← ENNReal.add_div,
     ENNReal.div_self (add_eq_zero.not.2 fun h => hs₀ h.1) (add_ne_top.2 ⟨hsμ, htμ⟩)]
 #align measure_theory.laverage_union_mem_open_segment MeasureTheory.laverage_union_mem_openSegment
+-/
 
+#print MeasureTheory.laverage_union_mem_segment /-
 theorem laverage_union_mem_segment (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
     (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) :
     ⨍⁻ x in s ∪ t, f x ∂μ ∈ [⨍⁻ x in s, f x ∂μ -[ℝ≥0∞] ⨍⁻ x in t, f x ∂μ] :=
@@ -199,7 +236,9 @@ theorem laverage_union_mem_segment (hd : AEDisjoint μ s t) (ht : NullMeasurable
     rw [← ENNReal.add_div,
       ENNReal.div_self (add_eq_zero.not.2 fun h => hs₀ h.1) (add_ne_top.2 ⟨hsμ, htμ⟩)]
 #align measure_theory.laverage_union_mem_segment MeasureTheory.laverage_union_mem_segment
+-/
 
+#print MeasureTheory.laverage_mem_openSegment_compl_self /-
 theorem laverage_mem_openSegment_compl_self [IsFiniteMeasure μ] (hs : NullMeasurableSet s μ)
     (hs₀ : μ s ≠ 0) (hsc₀ : μ (sᶜ) ≠ 0) :
     ⨍⁻ x, f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ x in s, f x ∂μ) (⨍⁻ x in sᶜ, f x ∂μ) := by
@@ -207,7 +246,9 @@ theorem laverage_mem_openSegment_compl_self [IsFiniteMeasure μ] (hs : NullMeasu
     laverage_union_mem_open_segment ae_disjoint_compl_right hs.compl hs₀ hsc₀ (measure_ne_top _ _)
       (measure_ne_top _ _)
 #align measure_theory.laverage_mem_open_segment_compl_self MeasureTheory.laverage_mem_openSegment_compl_self
+-/
 
+#print MeasureTheory.laverage_const /-
 @[simp]
 theorem laverage_const (μ : Measure α) [IsFiniteMeasure μ] [h : μ.ae.ne_bot] (c : ℝ≥0∞) :
     ⨍⁻ x, c ∂μ = c := by
@@ -215,21 +256,29 @@ theorem laverage_const (μ : Measure α) [IsFiniteMeasure μ] [h : μ.ae.ne_bot]
     div_eq_mul_inv, mul_assoc, ENNReal.mul_inv_cancel, mul_one, measure_ne_top μ univ, Ne.def,
     measure_univ_ne_zero, ae_ne_bot.1 h, not_false_iff]
 #align measure_theory.laverage_const MeasureTheory.laverage_const
+-/
 
-theorem set_laverage_const (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : ℝ≥0∞) : ⨍⁻ x in s, c ∂μ = c := by
+#print MeasureTheory.setLaverage_const /-
+theorem setLaverage_const (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : ℝ≥0∞) : ⨍⁻ x in s, c ∂μ = c := by
   simp only [set_laverage_eq, lintegral_const, measure.restrict_apply, MeasurableSet.univ,
     univ_inter, div_eq_mul_inv, mul_assoc, ENNReal.mul_inv_cancel hs₀ hs, mul_one]
-#align measure_theory.set_laverage_const MeasureTheory.set_laverage_const
+#align measure_theory.set_laverage_const MeasureTheory.setLaverage_const
+-/
 
+#print MeasureTheory.laverage_one /-
 @[simp]
 theorem laverage_one [IsFiniteMeasure μ] [h : μ.ae.ne_bot] : ⨍⁻ x, (1 : ℝ≥0∞) ∂μ = 1 :=
   laverage_const _ _
 #align measure_theory.laverage_one MeasureTheory.laverage_one
+-/
 
-theorem set_laverage_one (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) : ⨍⁻ x in s, (1 : ℝ≥0∞) ∂μ = 1 :=
-  set_laverage_const hs₀ hs _
-#align measure_theory.set_laverage_one MeasureTheory.set_laverage_one
+#print MeasureTheory.setLaverage_one /-
+theorem setLaverage_one (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) : ⨍⁻ x in s, (1 : ℝ≥0∞) ∂μ = 1 :=
+  setLaverage_const hs₀ hs _
+#align measure_theory.set_laverage_one MeasureTheory.setLaverage_one
+-/
 
+#print MeasureTheory.lintegral_laverage /-
 @[simp]
 theorem lintegral_laverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ≥0∞) :
     ∫⁻ x, ⨍⁻ a, f a ∂μ ∂μ = ∫⁻ x, f x ∂μ :=
@@ -240,11 +289,14 @@ theorem lintegral_laverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → 
     rw [lintegral_const, laverage_eq,
       ENNReal.div_mul_cancel (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)]
 #align measure_theory.lintegral_laverage MeasureTheory.lintegral_laverage
+-/
 
-theorem set_lintegral_set_laverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ≥0∞) (s : Set α) :
+#print MeasureTheory.setLintegral_setLaverage /-
+theorem setLintegral_setLaverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ≥0∞) (s : Set α) :
     ∫⁻ x in s, ⨍⁻ a in s, f a ∂μ ∂μ = ∫⁻ x in s, f x ∂μ :=
   lintegral_laverage _ _
-#align measure_theory.set_lintegral_set_laverage MeasureTheory.set_lintegral_set_laverage
+#align measure_theory.set_lintegral_set_laverage MeasureTheory.setLintegral_setLaverage
+-/
 
 end ENNReal
 
@@ -324,17 +376,17 @@ theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
 #align measure_theory.measure_smul_average MeasureTheory.measure_smul_average
 -/
 
-#print MeasureTheory.set_average_eq /-
-theorem set_average_eq (f : α → E) (s : Set α) :
+#print MeasureTheory.setAverage_eq /-
+theorem setAverage_eq (f : α → E) (s : Set α) :
     ⨍ x in s, f x ∂μ = (μ s).toReal⁻¹ • ∫ x in s, f x ∂μ := by rw [average_eq, restrict_apply_univ]
-#align measure_theory.set_average_eq MeasureTheory.set_average_eq
+#align measure_theory.set_average_eq MeasureTheory.setAverage_eq
 -/
 
-#print MeasureTheory.set_average_eq' /-
-theorem set_average_eq' (f : α → E) (s : Set α) :
+#print MeasureTheory.setAverage_eq' /-
+theorem setAverage_eq' (f : α → E) (s : Set α) :
     ⨍ x in s, f x ∂μ = ∫ x, f x ∂(μ s)⁻¹ • μ.restrict s := by
   simp only [average_eq', restrict_apply_univ]
-#align measure_theory.set_average_eq' MeasureTheory.set_average_eq'
+#align measure_theory.set_average_eq' MeasureTheory.setAverage_eq'
 -/
 
 variable {μ}
@@ -345,13 +397,17 @@ theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : ⨍ x, f x ∂μ = 
 #align measure_theory.average_congr MeasureTheory.average_congr
 -/
 
-theorem set_average_congr (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ := by
+#print MeasureTheory.setAverage_congr /-
+theorem setAverage_congr (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ := by
   simp only [set_average_eq, set_integral_congr_set_ae h, measure_congr h]
-#align measure_theory.set_average_congr MeasureTheory.set_average_congr
+#align measure_theory.set_average_congr MeasureTheory.setAverage_congr
+-/
 
-theorem set_average_congr_fun (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
+#print MeasureTheory.setAverage_congr_fun /-
+theorem setAverage_congr_fun (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
     ⨍ x in s, f x ∂μ = ⨍ x in s, g x ∂μ := by simp only [average_eq, set_integral_congr_ae hs h]
-#align measure_theory.set_average_congr_fun MeasureTheory.set_average_congr_fun
+#align measure_theory.set_average_congr_fun MeasureTheory.setAverage_congr_fun
+-/
 
 #print MeasureTheory.average_add_measure /-
 theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E}
@@ -373,11 +429,11 @@ theorem average_pair {f : α → E} {g : α → F} (hfi : Integrable f μ) (hgi
 #align measure_theory.average_pair MeasureTheory.average_pair
 -/
 
-#print MeasureTheory.measure_smul_set_average /-
-theorem measure_smul_set_average (f : α → E) {s : Set α} (h : μ s ≠ ∞) :
+#print MeasureTheory.measure_smul_setAverage /-
+theorem measure_smul_setAverage (f : α → E) {s : Set α} (h : μ s ≠ ∞) :
     (μ s).toReal • ⨍ x in s, f x ∂μ = ∫ x in s, f x ∂μ := by haveI := Fact.mk h.lt_top;
   rw [← measure_smul_average, restrict_apply_univ]
-#align measure_theory.measure_smul_set_average MeasureTheory.measure_smul_set_average
+#align measure_theory.measure_smul_set_average MeasureTheory.measure_smul_setAverage
 -/
 
 #print MeasureTheory.average_union /-
@@ -446,13 +502,13 @@ theorem average_const (μ : Measure α) [IsFiniteMeasure μ] [h : μ.ae.ne_bot]
 #align measure_theory.average_const MeasureTheory.average_const
 -/
 
-#print MeasureTheory.set_average_const /-
-theorem set_average_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : E) : ⨍ x in s, c ∂μ = c :=
+#print MeasureTheory.setAverage_const /-
+theorem setAverage_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : E) : ⨍ x in s, c ∂μ = c :=
   by
   simp only [set_average_eq, integral_const, measure.restrict_apply, MeasurableSet.univ, univ_inter,
     smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel hs₀ hs,
     ENNReal.one_toReal, one_smul]
-#align measure_theory.set_average_const MeasureTheory.set_average_const
+#align measure_theory.set_average_const MeasureTheory.setAverage_const
 -/
 
 #print MeasureTheory.integral_average /-
@@ -468,11 +524,11 @@ theorem integral_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :
 #align measure_theory.integral_average MeasureTheory.integral_average
 -/
 
-#print MeasureTheory.setAverage_setAverage /-
-theorem setAverage_setAverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) (s : Set α) :
+#print MeasureTheory.setIntegral_setAverage /-
+theorem setIntegral_setAverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) (s : Set α) :
     ∫ x in s, ⨍ a in s, f a ∂μ ∂μ = ∫ x in s, f x ∂μ :=
   integral_average _ _
-#align measure_theory.set_integral_set_average MeasureTheory.setAverage_setAverage
+#align measure_theory.set_integral_set_average MeasureTheory.setIntegral_setAverage
 -/
 
 #print MeasureTheory.integral_sub_average /-
@@ -502,12 +558,12 @@ theorem integral_average_sub [IsFiniteMeasure μ] (hf : Integrable f μ) :
 #align measure_theory.integral_average_sub MeasureTheory.integral_average_sub
 -/
 
-#print MeasureTheory.setAverage_setAverage_sub /-
-theorem setAverage_setAverage_sub (hs : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+#print MeasureTheory.setIntegral_setAverage_sub /-
+theorem setIntegral_setAverage_sub (hs : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
     ∫ x in s, ⨍ a in s, f a ∂μ - f x ∂μ = 0 :=
   haveI haveI : Fact (μ s < ∞) := ⟨lt_top_iff_ne_top.2 hs⟩
   integral_average_sub hf
-#align measure_theory.set_integral_set_average_sub MeasureTheory.setAverage_setAverage_sub
+#align measure_theory.set_integral_set_average_sub MeasureTheory.setIntegral_setAverage_sub
 -/
 
 end NormedAddCommGroup
@@ -532,6 +588,7 @@ theorem ofReal_setAverage {f : α → ℝ} (hf : IntegrableOn f s μ) (hf₀ : 0
 #align measure_theory.of_real_set_average MeasureTheory.ofReal_setAverage
 -/
 
+#print MeasureTheory.toReal_laverage /-
 theorem toReal_laverage {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf' : ∀ᵐ x ∂μ, f x ≠ ∞) :
     (⨍⁻ x, f x ∂μ).toReal = ⨍ x, (f x).toReal ∂μ :=
   by
@@ -541,12 +598,15 @@ theorem toReal_laverage {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf' :
     rw [average_eq, laverage_eq, smul_eq_mul, to_real_div, div_eq_inv_mul, ←
       integral_to_real hf (hf'.mono fun _ => lt_top_iff_ne_top.2)]
 #align measure_theory.to_real_laverage MeasureTheory.toReal_laverage
+-/
 
-theorem toReal_set_laverage {f : α → ℝ≥0∞} (hf : AEMeasurable f (μ.restrict s))
+#print MeasureTheory.toReal_setLaverage /-
+theorem toReal_setLaverage {f : α → ℝ≥0∞} (hf : AEMeasurable f (μ.restrict s))
     (hf' : ∀ᵐ x ∂μ.restrict s, f x ≠ ∞) :
     ((∫⁻ x in s, f x ∂μ) / μ s).toReal = ⨍ x in s, (f x).toReal ∂μ := by
   simpa [laverage_eq] using to_real_laverage hf hf'
-#align measure_theory.to_real_set_laverage MeasureTheory.toReal_set_laverage
+#align measure_theory.to_real_set_laverage MeasureTheory.toReal_setLaverage
+-/
 
 /-! ### First moment method -/
 
@@ -740,9 +800,10 @@ section FirstMomentEnnreal
 
 variable {N : Set α} {f : α → ℝ≥0∞}
 
+#print MeasureTheory.measure_le_setLaverage_pos /-
 /-- **First moment method**. A measurable function is smaller than its mean on a set of positive
 measure. -/
-theorem measure_le_set_laverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞)
+theorem measure_le_setLaverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞)
     (hf : AEMeasurable f (μ.restrict s)) : 0 < μ ({x ∈ s | f x ≤ ⨍⁻ a in s, f a ∂μ}) :=
   by
   obtain h | h := eq_or_ne (∫⁻ a in s, f a ∂μ) ∞
@@ -762,11 +823,13 @@ theorem measure_le_set_laverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞)
   rwa [← to_real_laverage hf, to_real_le_to_real hx (set_laverage_lt_top h).Ne] at hfx 
   · simp_rw [ae_iff, not_ne_iff]
     exact measure_eq_top_of_lintegral_ne_top hf h
-#align measure_theory.measure_le_set_laverage_pos MeasureTheory.measure_le_set_laverage_pos
+#align measure_theory.measure_le_set_laverage_pos MeasureTheory.measure_le_setLaverage_pos
+-/
 
+#print MeasureTheory.measure_setLaverage_le_pos /-
 /-- **First moment method**. A measurable function is greater than its mean on a set of positive
 measure. -/
-theorem measure_set_laverage_le_pos (hμ : μ s ≠ 0) (hs : NullMeasurableSet s μ)
+theorem measure_setLaverage_le_pos (hμ : μ s ≠ 0) (hs : NullMeasurableSet s μ)
     (hint : ∫⁻ a in s, f a ∂μ ≠ ∞) : 0 < μ ({x ∈ s | ⨍⁻ a in s, f a ∂μ ≤ f x}) :=
   by
   obtain hμ₁ | hμ₁ := eq_or_ne (μ s) ∞
@@ -787,22 +850,28 @@ theorem measure_set_laverage_le_pos (hμ : μ s ≠ 0) (hs : NullMeasurableSet s
   exact hfx.trans (hgf _)
   · simp_rw [ae_iff, not_ne_iff]
     exact measure_eq_top_of_lintegral_ne_top hg.ae_measurable hint
-#align measure_theory.measure_set_laverage_le_pos MeasureTheory.measure_set_laverage_le_pos
+#align measure_theory.measure_set_laverage_le_pos MeasureTheory.measure_setLaverage_le_pos
+-/
 
+#print MeasureTheory.exists_le_setLaverage /-
 /-- **First moment method**. The minimum of a measurable function is smaller than its mean. -/
-theorem exists_le_set_laverage (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : AEMeasurable f (μ.restrict s)) :
+theorem exists_le_setLaverage (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : AEMeasurable f (μ.restrict s)) :
     ∃ x ∈ s, f x ≤ ⨍⁻ a in s, f a ∂μ :=
-  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_le_set_laverage_pos hμ hμ₁ hf).ne'
+  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_le_setLaverage_pos hμ hμ₁ hf).ne'
   ⟨x, hx, h⟩
-#align measure_theory.exists_le_set_laverage MeasureTheory.exists_le_set_laverage
+#align measure_theory.exists_le_set_laverage MeasureTheory.exists_le_setLaverage
+-/
 
+#print MeasureTheory.exists_setLaverage_le /-
 /-- **First moment method**. The maximum of a measurable function is greater than its mean. -/
-theorem exists_set_laverage_le (hμ : μ s ≠ 0) (hs : NullMeasurableSet s μ)
+theorem exists_setLaverage_le (hμ : μ s ≠ 0) (hs : NullMeasurableSet s μ)
     (hint : ∫⁻ a in s, f a ∂μ ≠ ∞) : ∃ x ∈ s, ⨍⁻ a in s, f a ∂μ ≤ f x :=
-  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_set_laverage_le_pos hμ hs hint).ne'
+  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_setLaverage_le_pos hμ hs hint).ne'
   ⟨x, hx, h⟩
-#align measure_theory.exists_set_laverage_le MeasureTheory.exists_set_laverage_le
+#align measure_theory.exists_set_laverage_le MeasureTheory.exists_setLaverage_le
+-/
 
+#print MeasureTheory.measure_laverage_le_pos /-
 /-- **First moment method**. A measurable function is greater than its mean on a set of positive
 measure. -/
 theorem measure_laverage_le_pos (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞) :
@@ -810,14 +879,18 @@ theorem measure_laverage_le_pos (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠
   simpa [hint] using
     @measure_set_laverage_le_pos _ _ _ _ f (measure_univ_ne_zero.2 hμ) null_measurable_set_univ
 #align measure_theory.measure_laverage_le_pos MeasureTheory.measure_laverage_le_pos
+-/
 
+#print MeasureTheory.exists_laverage_le /-
 /-- **First moment method**. The maximum of a measurable function is greater than its mean. -/
 theorem exists_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞) : ∃ x, ⨍⁻ a, f a ∂μ ≤ f x :=
   let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_laverage_le_pos hμ hint).ne'
   ⟨x, hx⟩
 #align measure_theory.exists_laverage_le MeasureTheory.exists_laverage_le
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+#print MeasureTheory.exists_not_mem_null_laverage_le /-
 /-- **First moment method**. The maximum of a measurable function is greater than its mean, while
 avoiding a null set. -/
 theorem exists_not_mem_null_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a : α, f a ∂μ ≠ ∞) (hN : μ N = 0) :
@@ -828,11 +901,13 @@ theorem exists_not_mem_null_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a : α,
   obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
   exact ⟨x, hxN, hx⟩
 #align measure_theory.exists_not_mem_null_laverage_le MeasureTheory.exists_not_mem_null_laverage_le
+-/
 
 section FiniteMeasure
 
 variable [IsFiniteMeasure μ]
 
+#print MeasureTheory.measure_le_laverage_pos /-
 /-- **First moment method**. A measurable function is smaller than its mean on a set of positive
 measure. -/
 theorem measure_le_laverage_pos (hμ : μ ≠ 0) (hf : AEMeasurable f μ) :
@@ -840,14 +915,18 @@ theorem measure_le_laverage_pos (hμ : μ ≠ 0) (hf : AEMeasurable f μ) :
   simpa using
     measure_le_set_laverage_pos (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _) hf.restrict
 #align measure_theory.measure_le_laverage_pos MeasureTheory.measure_le_laverage_pos
+-/
 
+#print MeasureTheory.exists_le_laverage /-
 /-- **First moment method**. The minimum of a measurable function is smaller than its mean. -/
 theorem exists_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) : ∃ x, f x ≤ ⨍⁻ a, f a ∂μ :=
   let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_le_laverage_pos hμ hf).ne'
   ⟨x, hx⟩
 #align measure_theory.exists_le_laverage MeasureTheory.exists_le_laverage
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+#print MeasureTheory.exists_not_mem_null_le_laverage /-
 /-- **First moment method**. The minimum of a measurable function is smaller than its mean, while
 avoiding a null set. -/
 theorem exists_not_mem_null_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) (hN : μ N = 0) :
@@ -858,6 +937,7 @@ theorem exists_not_mem_null_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ
   obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
   exact ⟨x, hxN, hx⟩
 #align measure_theory.exists_not_mem_null_le_laverage MeasureTheory.exists_not_mem_null_le_laverage
+-/
 
 end FiniteMeasure
 
@@ -865,32 +945,41 @@ section ProbabilityMeasure
 
 variable [IsProbabilityMeasure μ]
 
+#print MeasureTheory.measure_le_lintegral_pos /-
 /-- **First moment method**. A measurable function is smaller than its integral on a set of
 positive measure. -/
 theorem measure_le_lintegral_pos (hf : AEMeasurable f μ) : 0 < μ {x | f x ≤ ∫⁻ a, f a ∂μ} := by
   simpa only [laverage_eq_lintegral] using
     measure_le_laverage_pos (is_probability_measure.ne_zero μ) hf
 #align measure_theory.measure_le_lintegral_pos MeasureTheory.measure_le_lintegral_pos
+-/
 
+#print MeasureTheory.measure_lintegral_le_pos /-
 /-- **First moment method**. A measurable function is greater than its integral on a set of
 positive measure. -/
 theorem measure_lintegral_le_pos (hint : ∫⁻ a, f a ∂μ ≠ ∞) : 0 < μ {x | ∫⁻ a, f a ∂μ ≤ f x} := by
   simpa only [laverage_eq_lintegral] using
     measure_laverage_le_pos (is_probability_measure.ne_zero μ) hint
 #align measure_theory.measure_lintegral_le_pos MeasureTheory.measure_lintegral_le_pos
+-/
 
+#print MeasureTheory.exists_le_lintegral /-
 /-- **First moment method**. The minimum of a measurable function is smaller than its integral. -/
 theorem exists_le_lintegral (hf : AEMeasurable f μ) : ∃ x, f x ≤ ∫⁻ a, f a ∂μ := by
   simpa only [laverage_eq_lintegral] using exists_le_laverage (is_probability_measure.ne_zero μ) hf
 #align measure_theory.exists_le_lintegral MeasureTheory.exists_le_lintegral
+-/
 
+#print MeasureTheory.exists_lintegral_le /-
 /-- **First moment method**. The maximum of a measurable function is greater than its integral. -/
 theorem exists_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) : ∃ x, ∫⁻ a, f a ∂μ ≤ f x := by
   simpa only [laverage_eq_lintegral] using
     exists_laverage_le (is_probability_measure.ne_zero μ) hint
 #align measure_theory.exists_lintegral_le MeasureTheory.exists_lintegral_le
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+#print MeasureTheory.exists_not_mem_null_le_lintegral /-
 /-- **First moment method**. The minimum of a measurable function is smaller than its integral,
 while avoiding a null set. -/
 theorem exists_not_mem_null_le_lintegral (hf : AEMeasurable f μ) (hN : μ N = 0) :
@@ -898,8 +987,10 @@ theorem exists_not_mem_null_le_lintegral (hf : AEMeasurable f μ) (hN : μ N = 0
   simpa only [laverage_eq_lintegral] using
     exists_not_mem_null_le_laverage (is_probability_measure.ne_zero μ) hf hN
 #align measure_theory.exists_not_mem_null_le_lintegral MeasureTheory.exists_not_mem_null_le_lintegral
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+#print MeasureTheory.exists_not_mem_null_lintegral_le /-
 /-- **First moment method**. The maximum of a measurable function is greater than its integral,
 while avoiding a null set. -/
 theorem exists_not_mem_null_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) (hN : μ N = 0) :
@@ -907,6 +998,7 @@ theorem exists_not_mem_null_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) (h
   simpa only [laverage_eq_lintegral] using
     exists_not_mem_null_laverage_le (is_probability_measure.ne_zero μ) hint hN
 #align measure_theory.exists_not_mem_null_lintegral_le MeasureTheory.exists_not_mem_null_lintegral_le
+-/
 
 end ProbabilityMeasure

bump to nightly-2023-07-16-17

mathlib commit https://github.com/leanprover-community/mathlib/commit/2fe465deb81bcd7ccafa065bb686888a82f15372

6db63fa6

Diff

@@ -455,6 +455,7 @@ theorem set_average_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞)
 #align measure_theory.set_average_const MeasureTheory.set_average_const
 -/
 
+#print MeasureTheory.integral_average /-
 @[simp]
 theorem integral_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :
     ∫ x, ⨍ a, f a ∂μ ∂μ = ∫ x, f x ∂μ :=
@@ -465,12 +466,16 @@ theorem integral_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :
     rw [integral_const, average_eq,
       smul_inv_smul₀ (to_real_ne_zero.2 ⟨measure_univ_ne_zero.2 hμ, measure_ne_top _ _⟩)]
 #align measure_theory.integral_average MeasureTheory.integral_average
+-/
 
-theorem set_integral_set_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) (s : Set α) :
+#print MeasureTheory.setAverage_setAverage /-
+theorem setAverage_setAverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) (s : Set α) :
     ∫ x in s, ⨍ a in s, f a ∂μ ∂μ = ∫ x in s, f x ∂μ :=
   integral_average _ _
-#align measure_theory.set_integral_set_average MeasureTheory.set_integral_set_average
+#align measure_theory.set_integral_set_average MeasureTheory.setAverage_setAverage
+-/
 
+#print MeasureTheory.integral_sub_average /-
 theorem integral_sub_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :
     ∫ x, f x - ⨍ a, f a ∂μ ∂μ = 0 :=
   by
@@ -480,26 +485,34 @@ theorem integral_sub_average (μ : Measure α) [IsFiniteMeasure μ] (f : α →
   convert h.add (integrable_const _)
   exact (sub_add_cancel _ _).symm
 #align measure_theory.integral_sub_average MeasureTheory.integral_sub_average
+-/
 
-theorem set_integral_sub_set_average (hs : μ s ≠ ∞) (f : α → E) :
+#print MeasureTheory.setAverage_sub_setAverage /-
+theorem setAverage_sub_setAverage (hs : μ s ≠ ∞) (f : α → E) :
     ∫ x in s, f x - ⨍ a in s, f a ∂μ ∂μ = 0 :=
   haveI haveI : Fact (μ s < ∞) := ⟨lt_top_iff_ne_top.2 hs⟩
   integral_sub_average _ _
-#align measure_theory.set_integral_sub_set_average MeasureTheory.set_integral_sub_set_average
+#align measure_theory.set_integral_sub_set_average MeasureTheory.setAverage_sub_setAverage
+-/
 
+#print MeasureTheory.integral_average_sub /-
 theorem integral_average_sub [IsFiniteMeasure μ] (hf : Integrable f μ) :
     ∫ x, ⨍ a, f a ∂μ - f x ∂μ = 0 := by
   rw [integral_sub (integrable_const _) hf, integral_average, sub_self]
 #align measure_theory.integral_average_sub MeasureTheory.integral_average_sub
+-/
 
-theorem set_integral_set_average_sub (hs : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+#print MeasureTheory.setAverage_setAverage_sub /-
+theorem setAverage_setAverage_sub (hs : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
     ∫ x in s, ⨍ a in s, f a ∂μ - f x ∂μ = 0 :=
   haveI haveI : Fact (μ s < ∞) := ⟨lt_top_iff_ne_top.2 hs⟩
   integral_average_sub hf
-#align measure_theory.set_integral_set_average_sub MeasureTheory.set_integral_set_average_sub
+#align measure_theory.set_integral_set_average_sub MeasureTheory.setAverage_setAverage_sub
+-/
 
 end NormedAddCommGroup
 
+#print MeasureTheory.ofReal_average /-
 theorem ofReal_average {f : α → ℝ} (hf : Integrable f μ) (hf₀ : 0 ≤ᵐ[μ] f) :
     ENNReal.ofReal (⨍ x, f x ∂μ) = (∫⁻ x, ENNReal.ofReal (f x) ∂μ) / μ univ :=
   by
@@ -510,11 +523,14 @@ theorem ofReal_average {f : α → ℝ} (hf : Integrable f μ) (hf₀ : 0 ≤ᵐ
       of_real_to_real (inv_ne_top.2 <| measure_univ_ne_zero.2 hμ),
       of_real_integral_eq_lintegral_of_real hf hf₀, ENNReal.div_eq_inv_mul]
 #align measure_theory.of_real_average MeasureTheory.ofReal_average
+-/
 
-theorem ofReal_set_average {f : α → ℝ} (hf : IntegrableOn f s μ) (hf₀ : 0 ≤ᵐ[μ.restrict s] f) :
+#print MeasureTheory.ofReal_setAverage /-
+theorem ofReal_setAverage {f : α → ℝ} (hf : IntegrableOn f s μ) (hf₀ : 0 ≤ᵐ[μ.restrict s] f) :
     ENNReal.ofReal (⨍ x in s, f x ∂μ) = (∫⁻ x in s, ENNReal.ofReal (f x) ∂μ) / μ s := by
   simpa using of_real_average hf hf₀
-#align measure_theory.of_real_set_average MeasureTheory.ofReal_set_average
+#align measure_theory.of_real_set_average MeasureTheory.ofReal_setAverage
+-/
 
 theorem toReal_laverage {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf' : ∀ᵐ x ∂μ, f x ≠ ∞) :
     (⨍⁻ x, f x ∂μ).toReal = ⨍ x, (f x).toReal ∂μ :=
@@ -539,9 +555,10 @@ section FirstMomentReal
 
 variable {N : Set α} {f : α → ℝ}
 
+#print MeasureTheory.measure_le_setAverage_pos /-
 /-- **First moment method**. An integrable function is smaller than its mean on a set of positive
 measure. -/
-theorem measure_le_set_average_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+theorem measure_le_setAverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
     0 < μ ({x ∈ s | f x ≤ ⨍ a in s, f a ∂μ}) :=
   by
   refine' pos_iff_ne_zero.2 fun H => _
@@ -559,33 +576,41 @@ theorem measure_le_set_average_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (h
   · rwa [pos_iff_ne_zero, inter_comm, ← diff_compl, ← diff_inter_self_eq_diff, measure_diff_null]
     refine' eq_bot_mono (measure_mono _) (measure_inter_eq_zero_of_restrict H)
     exact inter_subset_inter_left _ fun a ha => (sub_eq_zero.1 <| of_not_not ha).le
-#align measure_theory.measure_le_set_average_pos MeasureTheory.measure_le_set_average_pos
+#align measure_theory.measure_le_set_average_pos MeasureTheory.measure_le_setAverage_pos
+-/
 
+#print MeasureTheory.measure_setAverage_le_pos /-
 /-- **First moment method**. An integrable function is greater than its mean on a set of positive
 measure. -/
-theorem measure_set_average_le_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+theorem measure_setAverage_le_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
     0 < μ ({x ∈ s | ⨍ a in s, f a ∂μ ≤ f x}) := by
   simpa [integral_neg, neg_div] using measure_le_set_average_pos hμ hμ₁ hf.neg
-#align measure_theory.measure_set_average_le_pos MeasureTheory.measure_set_average_le_pos
+#align measure_theory.measure_set_average_le_pos MeasureTheory.measure_setAverage_le_pos
+-/
 
+#print MeasureTheory.exists_le_setAverage /-
 /-- **First moment method**. The minimum of an integrable function is smaller than its mean. -/
-theorem exists_le_set_average (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+theorem exists_le_setAverage (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
     ∃ x ∈ s, f x ≤ ⨍ a in s, f a ∂μ :=
-  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_le_set_average_pos hμ hμ₁ hf).ne'
+  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_le_setAverage_pos hμ hμ₁ hf).ne'
   ⟨x, hx, h⟩
-#align measure_theory.exists_le_set_average MeasureTheory.exists_le_set_average
+#align measure_theory.exists_le_set_average MeasureTheory.exists_le_setAverage
+-/
 
+#print MeasureTheory.exists_setAverage_le /-
 /-- **First moment method**. The maximum of an integrable function is greater than its mean. -/
-theorem exists_set_average_le (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+theorem exists_setAverage_le (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
     ∃ x ∈ s, ⨍ a in s, f a ∂μ ≤ f x :=
-  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_set_average_le_pos hμ hμ₁ hf).ne'
+  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_setAverage_le_pos hμ hμ₁ hf).ne'
   ⟨x, hx, h⟩
-#align measure_theory.exists_set_average_le MeasureTheory.exists_set_average_le
+#align measure_theory.exists_set_average_le MeasureTheory.exists_setAverage_le
+-/
 
 section FiniteMeasure
 
 variable [IsFiniteMeasure μ]
 
+#print MeasureTheory.measure_le_average_pos /-
 /-- **First moment method**. An integrable function is smaller than its mean on a set of positive
 measure. -/
 theorem measure_le_average_pos (hμ : μ ≠ 0) (hf : Integrable f μ) : 0 < μ {x | f x ≤ ⨍ a, f a ∂μ} :=
@@ -594,7 +619,9 @@ theorem measure_le_average_pos (hμ : μ ≠ 0) (hf : Integrable f μ) : 0 < μ
     measure_le_set_average_pos (measure.measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)
       hf.integrable_on
 #align measure_theory.measure_le_average_pos MeasureTheory.measure_le_average_pos
+-/
 
+#print MeasureTheory.measure_average_le_pos /-
 /-- **First moment method**. An integrable function is greater than its mean on a set of positive
 measure. -/
 theorem measure_average_le_pos (hμ : μ ≠ 0) (hf : Integrable f μ) : 0 < μ {x | ⨍ a, f a ∂μ ≤ f x} :=
@@ -603,20 +630,26 @@ theorem measure_average_le_pos (hμ : μ ≠ 0) (hf : Integrable f μ) : 0 < μ
     measure_set_average_le_pos (measure.measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)
       hf.integrable_on
 #align measure_theory.measure_average_le_pos MeasureTheory.measure_average_le_pos
+-/
 
+#print MeasureTheory.exists_le_average /-
 /-- **First moment method**. The minimum of an integrable function is smaller than its mean. -/
 theorem exists_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) : ∃ x, f x ≤ ⨍ a, f a ∂μ :=
   let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_le_average_pos hμ hf).ne'
   ⟨x, hx⟩
 #align measure_theory.exists_le_average MeasureTheory.exists_le_average
+-/
 
+#print MeasureTheory.exists_average_le /-
 /-- **First moment method**. The maximum of an integrable function is greater than its mean. -/
 theorem exists_average_le (hμ : μ ≠ 0) (hf : Integrable f μ) : ∃ x, ⨍ a, f a ∂μ ≤ f x :=
   let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_average_le_pos hμ hf).ne'
   ⟨x, hx⟩
 #align measure_theory.exists_average_le MeasureTheory.exists_average_le
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+#print MeasureTheory.exists_not_mem_null_le_average /-
 /-- **First moment method**. The minimum of an integrable function is smaller than its mean, while
 avoiding a null set. -/
 theorem exists_not_mem_null_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) (hN : μ N = 0) :
@@ -627,14 +660,17 @@ theorem exists_not_mem_null_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) (
   obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
   exact ⟨x, hxN, hx⟩
 #align measure_theory.exists_not_mem_null_le_average MeasureTheory.exists_not_mem_null_le_average
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+#print MeasureTheory.exists_not_mem_null_average_le /-
 /-- **First moment method**. The maximum of an integrable function is greater than its mean, while
 avoiding a null set. -/
 theorem exists_not_mem_null_average_le (hμ : μ ≠ 0) (hf : Integrable f μ) (hN : μ N = 0) :
     ∃ (x : _) (_ : x ∉ N), ⨍ a, f a ∂μ ≤ f x := by
   simpa [integral_neg, neg_div] using exists_not_mem_null_le_average hμ hf.neg hN
 #align measure_theory.exists_not_mem_null_average_le MeasureTheory.exists_not_mem_null_average_le
+-/
 
 end FiniteMeasure
 
@@ -642,31 +678,40 @@ section ProbabilityMeasure
 
 variable [IsProbabilityMeasure μ]
 
+#print MeasureTheory.measure_le_integral_pos /-
 /-- **First moment method**. An integrable function is smaller than its integral on a set of
 positive measure. -/
 theorem measure_le_integral_pos (hf : Integrable f μ) : 0 < μ {x | f x ≤ ∫ a, f a ∂μ} := by
   simpa only [average_eq_integral] using
     measure_le_average_pos (is_probability_measure.ne_zero μ) hf
 #align measure_theory.measure_le_integral_pos MeasureTheory.measure_le_integral_pos
+-/
 
+#print MeasureTheory.measure_integral_le_pos /-
 /-- **First moment method**. An integrable function is greater than its integral on a set of
 positive measure. -/
 theorem measure_integral_le_pos (hf : Integrable f μ) : 0 < μ {x | ∫ a, f a ∂μ ≤ f x} := by
   simpa only [average_eq_integral] using
     measure_average_le_pos (is_probability_measure.ne_zero μ) hf
 #align measure_theory.measure_integral_le_pos MeasureTheory.measure_integral_le_pos
+-/
 
+#print MeasureTheory.exists_le_integral /-
 /-- **First moment method**. The minimum of an integrable function is smaller than its integral. -/
 theorem exists_le_integral (hf : Integrable f μ) : ∃ x, f x ≤ ∫ a, f a ∂μ := by
   simpa only [average_eq_integral] using exists_le_average (is_probability_measure.ne_zero μ) hf
 #align measure_theory.exists_le_integral MeasureTheory.exists_le_integral
+-/
 
+#print MeasureTheory.exists_integral_le /-
 /-- **First moment method**. The maximum of an integrable function is greater than its integral. -/
 theorem exists_integral_le (hf : Integrable f μ) : ∃ x, ∫ a, f a ∂μ ≤ f x := by
   simpa only [average_eq_integral] using exists_average_le (is_probability_measure.ne_zero μ) hf
 #align measure_theory.exists_integral_le MeasureTheory.exists_integral_le
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+#print MeasureTheory.exists_not_mem_null_le_integral /-
 /-- **First moment method**. The minimum of an integrable function is smaller than its integral,
 while avoiding a null set. -/
 theorem exists_not_mem_null_le_integral (hf : Integrable f μ) (hN : μ N = 0) :
@@ -674,8 +719,10 @@ theorem exists_not_mem_null_le_integral (hf : Integrable f μ) (hN : μ N = 0) :
   simpa only [average_eq_integral] using
     exists_not_mem_null_le_average (is_probability_measure.ne_zero μ) hf hN
 #align measure_theory.exists_not_mem_null_le_integral MeasureTheory.exists_not_mem_null_le_integral
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+#print MeasureTheory.exists_not_mem_null_integral_le /-
 /-- **First moment method**. The maximum of an integrable function is greater than its integral,
 while avoiding a null set. -/
 theorem exists_not_mem_null_integral_le (hf : Integrable f μ) (hN : μ N = 0) :
@@ -683,6 +730,7 @@ theorem exists_not_mem_null_integral_le (hf : Integrable f μ) (hN : μ N = 0) :
   simpa only [average_eq_integral] using
     exists_not_mem_null_average_le (is_probability_measure.ne_zero μ) hf hN
 #align measure_theory.exists_not_mem_null_integral_le MeasureTheory.exists_not_mem_null_integral_le
+-/
 
 end ProbabilityMeasure

bump to nightly-2023-06-22-03

mathlib commit https://github.com/leanprover-community/mathlib/commit/6285167a053ad0990fc88e56c48ccd9fae6550eb

2424c6fa

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov, Yaël Dillies
 
 ! This file was ported from Lean 3 source module measure_theory.integral.average
-! leanprover-community/mathlib commit ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f
+! leanprover-community/mathlib commit c14c8fcde993801fca8946b0d80131a1a81d1520
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -24,6 +24,8 @@ measure, then the average of any function is equal to its integral.
 For the average on a set, we use `⨍ x in s, f x ∂μ` (notation for `⨍ x, f x ∂(μ.restrict s)`). For
 average w.r.t. the volume, one can omit `∂volume`.
 
+Both have a version for the Lebesgue integral rather than Bochner.
+
 We prove several version of the first moment method: An integrable function is below/above its
 average on a set of positive measure.
 
@@ -33,11 +35,6 @@ The average is defined as an integral over `(μ univ)⁻¹ • μ` so that all t
 integrals work for the average without modifications. For theorems that require integrability of a
 function, we provide a convenience lemma `measure_theory.integrable.to_average`.
 
-## TODO
-
-Provide the first moment method for the Lebesgue integral as well. A draft is available on branch
-`first_moment_lintegral`.
-
 ## Tags
 
 integral, center mass, average value
@@ -49,22 +46,208 @@ open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSp
 open scoped Topology BigOperators ENNReal Convex
 
 variable {α E F : Type _} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {μ : Measure α}
+  [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {μ ν : Measure α}
   {s t : Set α}
 
 /-!
 ### Average value of a function w.r.t. a measure
 
-The average value of a function `f` w.r.t. a measure `μ` (notation: `⨍ x, f x ∂μ`) is defined as
-`(μ univ).to_real⁻¹ • ∫ x, f x ∂μ`, so it is equal to zero if `f` is not integrable or if `μ` is an
-infinite measure. If `μ` is a probability measure, then the average of any function is equal to its
-integral.
-
+The (Bochner, Lebesgue) average value of a function `f` w.r.t. a measure `μ` (notation:
+`⨍ x, f x ∂μ`, `⨍⁻ x, f x ∂μ`) is defined as the (Bochner, Lebesgue) integral divided by the total
+measure, so it is equal to zero if `μ` is an infinite measure, and (typically) equal to infinity if
+`f` is not integrable. If `μ` is a probability measure, then the average of any function is equal to
+its integral.
 -/
 
 
 namespace MeasureTheory
 
+section ENNReal
+
+variable (μ) {f g : α → ℝ≥0∞}
+
+/-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. a measure `μ`, notation: `⨍⁻ x, f x ∂μ`.
+It is defined as `μ univ⁻¹ * ∫⁻ x, f x ∂μ`, so it is equal to zero if `μ` is an infinite measure. If
+`μ` is a probability measure, then the average of any function is equal to its integral.
+
+For the average on a set, use `⨍⁻ x in s, f x ∂μ` (defined as `⨍⁻ x, f x ∂(μ.restrict s)`). For
+average w.r.t. the volume, one can omit `∂volume`. -/
+noncomputable def laverage (f : α → ℝ≥0∞) :=
+  ∫⁻ x, f x ∂(μ univ)⁻¹ • μ
+#align measure_theory.laverage MeasureTheory.laverage
+
+notation3"⨍⁻ "(...)", "r:60:(scoped f => f)" ∂"μ:70 => laverage μ r
+
+notation3"⨍⁻ "(...)", "r:60:(scoped f => laverage volume f) => r
+
+notation3"⨍⁻ "(...)" in "s", "r:60:(scoped f => f)" ∂"μ:70 => laverage (Measure.restrict μ s) r
+
+notation3"⨍⁻ "(...)" in "s", "r:60:(scoped f => laverage Measure.restrict volume s f) => r
+
+@[simp]
+theorem laverage_zero : ⨍⁻ x, (0 : ℝ≥0∞) ∂μ = 0 := by rw [laverage, lintegral_zero]
+#align measure_theory.laverage_zero MeasureTheory.laverage_zero
+
+@[simp]
+theorem laverage_zero_measure (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂(0 : Measure α) = 0 := by simp [laverage]
+#align measure_theory.laverage_zero_measure MeasureTheory.laverage_zero_measure
+
+theorem laverage_eq' (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂(μ univ)⁻¹ • μ :=
+  rfl
+#align measure_theory.laverage_eq' MeasureTheory.laverage_eq'
+
+theorem laverage_eq (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = (∫⁻ x, f x ∂μ) / μ univ := by
+  rw [laverage_eq', lintegral_smul_measure, ENNReal.div_eq_inv_mul]
+#align measure_theory.laverage_eq MeasureTheory.laverage_eq
+
+theorem laverage_eq_lintegral [IsProbabilityMeasure μ] (f : α → ℝ≥0∞) :
+    ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ := by rw [laverage, measure_univ, inv_one, one_smul]
+#align measure_theory.laverage_eq_lintegral MeasureTheory.laverage_eq_lintegral
+
+@[simp]
+theorem measure_mul_laverage [IsFiniteMeasure μ] (f : α → ℝ≥0∞) :
+    μ univ * ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ :=
+  by
+  cases' eq_or_ne μ 0 with hμ hμ
+  · rw [hμ, lintegral_zero_measure, laverage_zero_measure, MulZeroClass.mul_zero]
+  · rw [laverage_eq, ENNReal.mul_div_cancel' (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)]
+#align measure_theory.measure_mul_laverage MeasureTheory.measure_mul_laverage
+
+theorem set_laverage_eq (f : α → ℝ≥0∞) (s : Set α) :
+    ⨍⁻ x in s, f x ∂μ = (∫⁻ x in s, f x ∂μ) / μ s := by rw [laverage_eq, restrict_apply_univ]
+#align measure_theory.set_laverage_eq MeasureTheory.set_laverage_eq
+
+theorem set_laverage_eq' (f : α → ℝ≥0∞) (s : Set α) :
+    ⨍⁻ x in s, f x ∂μ = ∫⁻ x, f x ∂(μ s)⁻¹ • μ.restrict s := by
+  simp only [laverage_eq', restrict_apply_univ]
+#align measure_theory.set_laverage_eq' MeasureTheory.set_laverage_eq'
+
+variable {μ}
+
+theorem laverage_congr {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : ⨍⁻ x, f x ∂μ = ⨍⁻ x, g x ∂μ := by
+  simp only [laverage_eq, lintegral_congr_ae h]
+#align measure_theory.laverage_congr MeasureTheory.laverage_congr
+
+theorem set_laverage_congr (h : s =ᵐ[μ] t) : ⨍⁻ x in s, f x ∂μ = ⨍⁻ x in t, f x ∂μ := by
+  simp only [set_laverage_eq, set_lintegral_congr h, measure_congr h]
+#align measure_theory.set_laverage_congr MeasureTheory.set_laverage_congr
+
+theorem set_laverage_congr_fun (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
+    ⨍⁻ x in s, f x ∂μ = ⨍⁻ x in s, g x ∂μ := by
+  simp only [laverage_eq, set_lintegral_congr_fun hs h]
+#align measure_theory.set_laverage_congr_fun MeasureTheory.set_laverage_congr_fun
+
+theorem laverage_lt_top (hf : ∫⁻ x, f x ∂μ ≠ ∞) : ⨍⁻ x, f x ∂μ < ∞ :=
+  by
+  obtain rfl | hμ := eq_or_ne μ 0
+  · simp
+  · rw [laverage_eq]
+    exact div_lt_top hf (measure_univ_ne_zero.2 hμ)
+#align measure_theory.laverage_lt_top MeasureTheory.laverage_lt_top
+
+theorem set_laverage_lt_top : ∫⁻ x in s, f x ∂μ ≠ ∞ → ⨍⁻ x in s, f x ∂μ < ∞ :=
+  laverage_lt_top
+#align measure_theory.set_laverage_lt_top MeasureTheory.set_laverage_lt_top
+
+theorem laverage_add_measure :
+    ⨍⁻ x, f x ∂(μ + ν) =
+      μ univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂μ + ν univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂ν :=
+  by
+  by_cases hμ : is_finite_measure μ; swap
+  · rw [not_is_finite_measure_iff] at hμ 
+    simp [laverage_eq, hμ]
+  by_cases hν : is_finite_measure ν; swap
+  · rw [not_is_finite_measure_iff] at hν 
+    simp [laverage_eq, hν]
+  haveI := hμ; haveI := hν
+  simp only [← ENNReal.mul_div_right_comm, measure_mul_laverage, ← ENNReal.add_div, ←
+    lintegral_add_measure, ← measure.add_apply, ← laverage_eq]
+#align measure_theory.laverage_add_measure MeasureTheory.laverage_add_measure
+
+theorem measure_mul_set_laverage (f : α → ℝ≥0∞) (h : μ s ≠ ∞) :
+    μ s * ⨍⁻ x in s, f x ∂μ = ∫⁻ x in s, f x ∂μ := by haveI := Fact.mk h.lt_top;
+  rw [← measure_mul_laverage, restrict_apply_univ]
+#align measure_theory.measure_mul_set_laverage MeasureTheory.measure_mul_set_laverage
+
+theorem laverage_union (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ) :
+    ⨍⁻ x in s ∪ t, f x ∂μ =
+      μ s / (μ s + μ t) * ⨍⁻ x in s, f x ∂μ + μ t / (μ s + μ t) * ⨍⁻ x in t, f x ∂μ :=
+  by rw [restrict_union₀ hd ht, laverage_add_measure, restrict_apply_univ, restrict_apply_univ]
+#align measure_theory.laverage_union MeasureTheory.laverage_union
+
+theorem laverage_union_mem_openSegment (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
+    (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) :
+    ⨍⁻ x in s ∪ t, f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ x in s, f x ∂μ) (⨍⁻ x in t, f x ∂μ) :=
+  by
+  refine'
+    ⟨μ s / (μ s + μ t), μ t / (μ s + μ t), ENNReal.div_pos hs₀ <| add_ne_top.2 ⟨hsμ, htμ⟩,
+      ENNReal.div_pos ht₀ <| add_ne_top.2 ⟨hsμ, htμ⟩, _, (laverage_union hd ht).symm⟩
+  rw [← ENNReal.add_div,
+    ENNReal.div_self (add_eq_zero.not.2 fun h => hs₀ h.1) (add_ne_top.2 ⟨hsμ, htμ⟩)]
+#align measure_theory.laverage_union_mem_open_segment MeasureTheory.laverage_union_mem_openSegment
+
+theorem laverage_union_mem_segment (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
+    (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) :
+    ⨍⁻ x in s ∪ t, f x ∂μ ∈ [⨍⁻ x in s, f x ∂μ -[ℝ≥0∞] ⨍⁻ x in t, f x ∂μ] :=
+  by
+  by_cases hs₀ : μ s = 0
+  · rw [← ae_eq_empty] at hs₀ 
+    rw [restrict_congr_set (hs₀.union eventually_eq.rfl), empty_union]
+    exact right_mem_segment _ _ _
+  · refine'
+      ⟨μ s / (μ s + μ t), μ t / (μ s + μ t), zero_le _, zero_le _, _, (laverage_union hd ht).symm⟩
+    rw [← ENNReal.add_div,
+      ENNReal.div_self (add_eq_zero.not.2 fun h => hs₀ h.1) (add_ne_top.2 ⟨hsμ, htμ⟩)]
+#align measure_theory.laverage_union_mem_segment MeasureTheory.laverage_union_mem_segment
+
+theorem laverage_mem_openSegment_compl_self [IsFiniteMeasure μ] (hs : NullMeasurableSet s μ)
+    (hs₀ : μ s ≠ 0) (hsc₀ : μ (sᶜ) ≠ 0) :
+    ⨍⁻ x, f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ x in s, f x ∂μ) (⨍⁻ x in sᶜ, f x ∂μ) := by
+  simpa only [union_compl_self, restrict_univ] using
+    laverage_union_mem_open_segment ae_disjoint_compl_right hs.compl hs₀ hsc₀ (measure_ne_top _ _)
+      (measure_ne_top _ _)
+#align measure_theory.laverage_mem_open_segment_compl_self MeasureTheory.laverage_mem_openSegment_compl_self
+
+@[simp]
+theorem laverage_const (μ : Measure α) [IsFiniteMeasure μ] [h : μ.ae.ne_bot] (c : ℝ≥0∞) :
+    ⨍⁻ x, c ∂μ = c := by
+  simp only [laverage_eq, lintegral_const, measure.restrict_apply, MeasurableSet.univ, univ_inter,
+    div_eq_mul_inv, mul_assoc, ENNReal.mul_inv_cancel, mul_one, measure_ne_top μ univ, Ne.def,
+    measure_univ_ne_zero, ae_ne_bot.1 h, not_false_iff]
+#align measure_theory.laverage_const MeasureTheory.laverage_const
+
+theorem set_laverage_const (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : ℝ≥0∞) : ⨍⁻ x in s, c ∂μ = c := by
+  simp only [set_laverage_eq, lintegral_const, measure.restrict_apply, MeasurableSet.univ,
+    univ_inter, div_eq_mul_inv, mul_assoc, ENNReal.mul_inv_cancel hs₀ hs, mul_one]
+#align measure_theory.set_laverage_const MeasureTheory.set_laverage_const
+
+@[simp]
+theorem laverage_one [IsFiniteMeasure μ] [h : μ.ae.ne_bot] : ⨍⁻ x, (1 : ℝ≥0∞) ∂μ = 1 :=
+  laverage_const _ _
+#align measure_theory.laverage_one MeasureTheory.laverage_one
+
+theorem set_laverage_one (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) : ⨍⁻ x in s, (1 : ℝ≥0∞) ∂μ = 1 :=
+  set_laverage_const hs₀ hs _
+#align measure_theory.set_laverage_one MeasureTheory.set_laverage_one
+
+@[simp]
+theorem lintegral_laverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ≥0∞) :
+    ∫⁻ x, ⨍⁻ a, f a ∂μ ∂μ = ∫⁻ x, f x ∂μ :=
+  by
+  obtain rfl | hμ := eq_or_ne μ 0
+  · simp
+  ·
+    rw [lintegral_const, laverage_eq,
+      ENNReal.div_mul_cancel (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)]
+#align measure_theory.lintegral_laverage MeasureTheory.lintegral_laverage
+
+theorem set_lintegral_set_laverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ≥0∞) (s : Set α) :
+    ∫⁻ x in s, ⨍⁻ a in s, f a ∂μ ∂μ = ∫⁻ x in s, f x ∂μ :=
+  lintegral_laverage _ _
+#align measure_theory.set_lintegral_set_laverage MeasureTheory.set_lintegral_set_laverage
+
+end ENNReal
+
 section NormedAddCommGroup
 
 variable (μ) {f g : α → E}
@@ -162,9 +345,13 @@ theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : ⨍ x, f x ∂μ = 
 #align measure_theory.average_congr MeasureTheory.average_congr
 -/
 
-theorem set_average_congr_set_ae (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ := by
+theorem set_average_congr (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ := by
   simp only [set_average_eq, set_integral_congr_set_ae h, measure_congr h]
-#align measure_theory.set_average_congr_set_ae MeasureTheory.set_average_congr_set_ae
+#align measure_theory.set_average_congr MeasureTheory.set_average_congr
+
+theorem set_average_congr_fun (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
+    ⨍ x in s, f x ∂μ = ⨍ x in s, g x ∂μ := by simp only [average_eq, set_integral_congr_ae hs h]
+#align measure_theory.set_average_congr_fun MeasureTheory.set_average_congr_fun
 
 #print MeasureTheory.average_add_measure /-
 theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E}
@@ -250,7 +437,8 @@ theorem average_mem_openSegment_compl_self [IsFiniteMeasure μ] {f : α → E} {
 
 #print MeasureTheory.average_const /-
 @[simp]
-theorem average_const [IsFiniteMeasure μ] [h : μ.ae.ne_bot] (c : E) : ⨍ x, c ∂μ = c := by
+theorem average_const (μ : Measure α) [IsFiniteMeasure μ] [h : μ.ae.ne_bot] (c : E) :
+    ⨍ x, c ∂μ = c := by
   simp only [average_eq, integral_const, measure.restrict_apply, MeasurableSet.univ, one_smul,
     univ_inter, smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel,
     measure_ne_top μ univ, Ne.def, measure_univ_eq_zero, ae_ne_bot.1 h, not_false_iff,
@@ -328,26 +516,26 @@ theorem ofReal_set_average {f : α → ℝ} (hf : IntegrableOn f s μ) (hf₀ :
   simpa using of_real_average hf hf₀
 #align measure_theory.of_real_set_average MeasureTheory.ofReal_set_average
 
-theorem average_toReal {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf' : ∀ᵐ x ∂μ, f x ≠ ∞) :
-    ⨍ x, (f x).toReal ∂μ = ((∫⁻ x, f x ∂μ) / μ univ).toReal :=
+theorem toReal_laverage {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf' : ∀ᵐ x ∂μ, f x ≠ ∞) :
+    (⨍⁻ x, f x ∂μ).toReal = ⨍ x, (f x).toReal ∂μ :=
   by
   obtain rfl | hμ := eq_or_ne μ 0
   · simp
   ·
-    rw [average_eq, smul_eq_mul, to_real_div, ←
-      integral_to_real hf (hf'.mono fun _ => lt_top_iff_ne_top.2), div_eq_inv_mul]
-#align measure_theory.average_to_real MeasureTheory.average_toReal
+    rw [average_eq, laverage_eq, smul_eq_mul, to_real_div, div_eq_inv_mul, ←
+      integral_to_real hf (hf'.mono fun _ => lt_top_iff_ne_top.2)]
+#align measure_theory.to_real_laverage MeasureTheory.toReal_laverage
 
-theorem set_average_toReal {f : α → ℝ≥0∞} (hf : AEMeasurable f (μ.restrict s))
+theorem toReal_set_laverage {f : α → ℝ≥0∞} (hf : AEMeasurable f (μ.restrict s))
     (hf' : ∀ᵐ x ∂μ.restrict s, f x ≠ ∞) :
-    ⨍ x in s, (f x).toReal ∂μ = ((∫⁻ x in s, f x ∂μ) / μ s).toReal := by
-  simpa using average_to_real hf hf'
-#align measure_theory.set_average_to_real MeasureTheory.set_average_toReal
+    ((∫⁻ x in s, f x ∂μ) / μ s).toReal = ⨍ x in s, (f x).toReal ∂μ := by
+  simpa [laverage_eq] using to_real_laverage hf hf'
+#align measure_theory.to_real_set_laverage MeasureTheory.toReal_set_laverage
 
 /-! ### First moment method -/
 
 
-section FirstMoment
+section FirstMomentReal
 
 variable {N : Set α} {f : α → ℝ}
 
@@ -498,7 +686,183 @@ theorem exists_not_mem_null_integral_le (hf : Integrable f μ) (hN : μ N = 0) :
 
 end ProbabilityMeasure
 
-end FirstMoment
+end FirstMomentReal
+
+section FirstMomentEnnreal
+
+variable {N : Set α} {f : α → ℝ≥0∞}
+
+/-- **First moment method**. A measurable function is smaller than its mean on a set of positive
+measure. -/
+theorem measure_le_set_laverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞)
+    (hf : AEMeasurable f (μ.restrict s)) : 0 < μ ({x ∈ s | f x ≤ ⨍⁻ a in s, f a ∂μ}) :=
+  by
+  obtain h | h := eq_or_ne (∫⁻ a in s, f a ∂μ) ∞
+  · simpa [mul_top, hμ₁, laverage, h, top_div_of_ne_top hμ₁, pos_iff_ne_zero] using hμ
+  have := measure_le_set_average_pos hμ hμ₁ (integrable_to_real_of_lintegral_ne_top hf h)
+  rw [← set_of_inter_eq_sep, ←
+    measure.restrict_apply₀
+      (hf.ae_strongly_measurable.null_measurable_set_le ae_strongly_measurable_const)]
+  rw [← set_of_inter_eq_sep, ←
+    measure.restrict_apply₀
+      (hf.ennreal_to_real.ae_strongly_measurable.null_measurable_set_le
+        ae_strongly_measurable_const),
+    ← measure_diff_null (measure_eq_top_of_lintegral_ne_top hf h)] at this 
+  refine' this.trans_le (measure_mono _)
+  rintro x ⟨hfx, hx⟩
+  dsimp at hfx 
+  rwa [← to_real_laverage hf, to_real_le_to_real hx (set_laverage_lt_top h).Ne] at hfx 
+  · simp_rw [ae_iff, not_ne_iff]
+    exact measure_eq_top_of_lintegral_ne_top hf h
+#align measure_theory.measure_le_set_laverage_pos MeasureTheory.measure_le_set_laverage_pos
+
+/-- **First moment method**. A measurable function is greater than its mean on a set of positive
+measure. -/
+theorem measure_set_laverage_le_pos (hμ : μ s ≠ 0) (hs : NullMeasurableSet s μ)
+    (hint : ∫⁻ a in s, f a ∂μ ≠ ∞) : 0 < μ ({x ∈ s | ⨍⁻ a in s, f a ∂μ ≤ f x}) :=
+  by
+  obtain hμ₁ | hμ₁ := eq_or_ne (μ s) ∞
+  · simp [set_laverage_eq, hμ₁]
+  obtain ⟨g, hg, hgf, hfg⟩ := exists_measurable_le_lintegral_eq (μ.restrict s) f
+  have hfg' : ⨍⁻ a in s, f a ∂μ = ⨍⁻ a in s, g a ∂μ := by simp_rw [laverage_eq, hfg]
+  rw [hfg] at hint 
+  have :=
+    measure_set_average_le_pos hμ hμ₁ (integrable_to_real_of_lintegral_ne_top hg.ae_measurable hint)
+  simp_rw [← set_of_inter_eq_sep, ← measure.restrict_apply₀' hs, hfg']
+  rw [← set_of_inter_eq_sep, ← measure.restrict_apply₀' hs, ←
+    measure_diff_null (measure_eq_top_of_lintegral_ne_top hg.ae_measurable hint)] at this 
+  refine' this.trans_le (measure_mono _)
+  rintro x ⟨hfx, hx⟩
+  dsimp at hfx 
+  rw [← to_real_laverage hg.ae_measurable, to_real_le_to_real (set_laverage_lt_top hint).Ne hx] at
+    hfx 
+  exact hfx.trans (hgf _)
+  · simp_rw [ae_iff, not_ne_iff]
+    exact measure_eq_top_of_lintegral_ne_top hg.ae_measurable hint
+#align measure_theory.measure_set_laverage_le_pos MeasureTheory.measure_set_laverage_le_pos
+
+/-- **First moment method**. The minimum of a measurable function is smaller than its mean. -/
+theorem exists_le_set_laverage (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : AEMeasurable f (μ.restrict s)) :
+    ∃ x ∈ s, f x ≤ ⨍⁻ a in s, f a ∂μ :=
+  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_le_set_laverage_pos hμ hμ₁ hf).ne'
+  ⟨x, hx, h⟩
+#align measure_theory.exists_le_set_laverage MeasureTheory.exists_le_set_laverage
+
+/-- **First moment method**. The maximum of a measurable function is greater than its mean. -/
+theorem exists_set_laverage_le (hμ : μ s ≠ 0) (hs : NullMeasurableSet s μ)
+    (hint : ∫⁻ a in s, f a ∂μ ≠ ∞) : ∃ x ∈ s, ⨍⁻ a in s, f a ∂μ ≤ f x :=
+  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_set_laverage_le_pos hμ hs hint).ne'
+  ⟨x, hx, h⟩
+#align measure_theory.exists_set_laverage_le MeasureTheory.exists_set_laverage_le
+
+/-- **First moment method**. A measurable function is greater than its mean on a set of positive
+measure. -/
+theorem measure_laverage_le_pos (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞) :
+    0 < μ {x | ⨍⁻ a, f a ∂μ ≤ f x} := by
+  simpa [hint] using
+    @measure_set_laverage_le_pos _ _ _ _ f (measure_univ_ne_zero.2 hμ) null_measurable_set_univ
+#align measure_theory.measure_laverage_le_pos MeasureTheory.measure_laverage_le_pos
+
+/-- **First moment method**. The maximum of a measurable function is greater than its mean. -/
+theorem exists_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞) : ∃ x, ⨍⁻ a, f a ∂μ ≤ f x :=
+  let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_laverage_le_pos hμ hint).ne'
+  ⟨x, hx⟩
+#align measure_theory.exists_laverage_le MeasureTheory.exists_laverage_le
+
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/-- **First moment method**. The maximum of a measurable function is greater than its mean, while
+avoiding a null set. -/
+theorem exists_not_mem_null_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a : α, f a ∂μ ≠ ∞) (hN : μ N = 0) :
+    ∃ (x : _) (_ : x ∉ N), ⨍⁻ a, f a ∂μ ≤ f x :=
+  by
+  have := measure_laverage_le_pos hμ hint
+  rw [← measure_diff_null hN] at this 
+  obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
+  exact ⟨x, hxN, hx⟩
+#align measure_theory.exists_not_mem_null_laverage_le MeasureTheory.exists_not_mem_null_laverage_le
+
+section FiniteMeasure
+
+variable [IsFiniteMeasure μ]
+
+/-- **First moment method**. A measurable function is smaller than its mean on a set of positive
+measure. -/
+theorem measure_le_laverage_pos (hμ : μ ≠ 0) (hf : AEMeasurable f μ) :
+    0 < μ {x | f x ≤ ⨍⁻ a, f a ∂μ} := by
+  simpa using
+    measure_le_set_laverage_pos (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _) hf.restrict
+#align measure_theory.measure_le_laverage_pos MeasureTheory.measure_le_laverage_pos
+
+/-- **First moment method**. The minimum of a measurable function is smaller than its mean. -/
+theorem exists_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) : ∃ x, f x ≤ ⨍⁻ a, f a ∂μ :=
+  let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_le_laverage_pos hμ hf).ne'
+  ⟨x, hx⟩
+#align measure_theory.exists_le_laverage MeasureTheory.exists_le_laverage
+
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/-- **First moment method**. The minimum of a measurable function is smaller than its mean, while
+avoiding a null set. -/
+theorem exists_not_mem_null_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) (hN : μ N = 0) :
+    ∃ (x : _) (_ : x ∉ N), f x ≤ ⨍⁻ a, f a ∂μ :=
+  by
+  have := measure_le_laverage_pos hμ hf
+  rw [← measure_diff_null hN] at this 
+  obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
+  exact ⟨x, hxN, hx⟩
+#align measure_theory.exists_not_mem_null_le_laverage MeasureTheory.exists_not_mem_null_le_laverage
+
+end FiniteMeasure
+
+section ProbabilityMeasure
+
+variable [IsProbabilityMeasure μ]
+
+/-- **First moment method**. A measurable function is smaller than its integral on a set of
+positive measure. -/
+theorem measure_le_lintegral_pos (hf : AEMeasurable f μ) : 0 < μ {x | f x ≤ ∫⁻ a, f a ∂μ} := by
+  simpa only [laverage_eq_lintegral] using
+    measure_le_laverage_pos (is_probability_measure.ne_zero μ) hf
+#align measure_theory.measure_le_lintegral_pos MeasureTheory.measure_le_lintegral_pos
+
+/-- **First moment method**. A measurable function is greater than its integral on a set of
+positive measure. -/
+theorem measure_lintegral_le_pos (hint : ∫⁻ a, f a ∂μ ≠ ∞) : 0 < μ {x | ∫⁻ a, f a ∂μ ≤ f x} := by
+  simpa only [laverage_eq_lintegral] using
+    measure_laverage_le_pos (is_probability_measure.ne_zero μ) hint
+#align measure_theory.measure_lintegral_le_pos MeasureTheory.measure_lintegral_le_pos
+
+/-- **First moment method**. The minimum of a measurable function is smaller than its integral. -/
+theorem exists_le_lintegral (hf : AEMeasurable f μ) : ∃ x, f x ≤ ∫⁻ a, f a ∂μ := by
+  simpa only [laverage_eq_lintegral] using exists_le_laverage (is_probability_measure.ne_zero μ) hf
+#align measure_theory.exists_le_lintegral MeasureTheory.exists_le_lintegral
+
+/-- **First moment method**. The maximum of a measurable function is greater than its integral. -/
+theorem exists_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) : ∃ x, ∫⁻ a, f a ∂μ ≤ f x := by
+  simpa only [laverage_eq_lintegral] using
+    exists_laverage_le (is_probability_measure.ne_zero μ) hint
+#align measure_theory.exists_lintegral_le MeasureTheory.exists_lintegral_le
+
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/-- **First moment method**. The minimum of a measurable function is smaller than its integral,
+while avoiding a null set. -/
+theorem exists_not_mem_null_le_lintegral (hf : AEMeasurable f μ) (hN : μ N = 0) :
+    ∃ (x : _) (_ : x ∉ N), f x ≤ ∫⁻ a, f a ∂μ := by
+  simpa only [laverage_eq_lintegral] using
+    exists_not_mem_null_le_laverage (is_probability_measure.ne_zero μ) hf hN
+#align measure_theory.exists_not_mem_null_le_lintegral MeasureTheory.exists_not_mem_null_le_lintegral
+
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/-- **First moment method**. The maximum of a measurable function is greater than its integral,
+while avoiding a null set. -/
+theorem exists_not_mem_null_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) (hN : μ N = 0) :
+    ∃ (x : _) (_ : x ∉ N), ∫⁻ a, f a ∂μ ≤ f x := by
+  simpa only [laverage_eq_lintegral] using
+    exists_not_mem_null_laverage_le (is_probability_measure.ne_zero μ) hint hN
+#align measure_theory.exists_not_mem_null_lintegral_le MeasureTheory.exists_not_mem_null_lintegral_le
+
+end ProbabilityMeasure
+
+end FirstMomentEnnreal
 
 end MeasureTheory

bump to nightly-2023-06-13-21

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

829520ac

Diff

@@ -69,8 +69,6 @@ section NormedAddCommGroup
 
 variable (μ) {f g : α → E}
 
-include m0
-
 #print MeasureTheory.average /-
 /-- Average value of a function `f` w.r.t. a measure `μ`, notation: `⨍ x, f x ∂μ`. It is defined as
 `(μ univ).to_real⁻¹ • ∫ x, f x ∂μ`, so it is equal to zero if `f` is not integrable or if `μ` is an
@@ -84,44 +82,53 @@ noncomputable def average (f : α → E) :=
 #align measure_theory.average MeasureTheory.average
 -/
 
--- mathport name: «expr⨍ , ∂ »
 notation3"⨍ "(...)", "r:60:(scoped f => f)" ∂"μ:70 => average μ r
 
--- mathport name: «expr⨍ , »
 notation3"⨍ "(...)", "r:60:(scoped f => average volume f) => r
 
--- mathport name: «expr⨍ in , ∂ »
 notation3"⨍ "(...)" in "s", "r:60:(scoped f => f)" ∂"μ:70 => average (Measure.restrict μ s) r
 
--- mathport name: «expr⨍ in , »
 notation3"⨍ "(...)" in "s", "r:60:(scoped f => average Measure.restrict volume s f) => r
 
+#print MeasureTheory.average_zero /-
 @[simp]
 theorem average_zero : ⨍ x, (0 : E) ∂μ = 0 := by rw [average, integral_zero]
 #align measure_theory.average_zero MeasureTheory.average_zero
+-/
 
+#print MeasureTheory.average_zero_measure /-
 @[simp]
 theorem average_zero_measure (f : α → E) : ⨍ x, f x ∂(0 : Measure α) = 0 := by
   rw [average, smul_zero, integral_zero_measure]
 #align measure_theory.average_zero_measure MeasureTheory.average_zero_measure
+-/
 
+#print MeasureTheory.average_neg /-
 @[simp]
 theorem average_neg (f : α → E) : ⨍ x, -f x ∂μ = -⨍ x, f x ∂μ :=
   integral_neg f
 #align measure_theory.average_neg MeasureTheory.average_neg
+-/
 
+#print MeasureTheory.average_eq' /-
 theorem average_eq' (f : α → E) : ⨍ x, f x ∂μ = ∫ x, f x ∂(μ univ)⁻¹ • μ :=
   rfl
 #align measure_theory.average_eq' MeasureTheory.average_eq'
+-/
 
+#print MeasureTheory.average_eq /-
 theorem average_eq (f : α → E) : ⨍ x, f x ∂μ = (μ univ).toReal⁻¹ • ∫ x, f x ∂μ := by
   rw [average_eq', integral_smul_measure, ENNReal.toReal_inv]
 #align measure_theory.average_eq MeasureTheory.average_eq
+-/
 
+#print MeasureTheory.average_eq_integral /-
 theorem average_eq_integral [IsProbabilityMeasure μ] (f : α → E) : ⨍ x, f x ∂μ = ∫ x, f x ∂μ := by
   rw [average, measure_univ, inv_one, one_smul]
 #align measure_theory.average_eq_integral MeasureTheory.average_eq_integral
+-/
 
+#print MeasureTheory.measure_smul_average /-
 @[simp]
 theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
     (μ univ).toReal • ⨍ x, f x ∂μ = ∫ x, f x ∂μ :=
@@ -132,26 +139,34 @@ theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
     refine' (ENNReal.toReal_pos _ <| measure_ne_top _ _).ne'
     rwa [Ne.def, measure_univ_eq_zero]
 #align measure_theory.measure_smul_average MeasureTheory.measure_smul_average
+-/
 
+#print MeasureTheory.set_average_eq /-
 theorem set_average_eq (f : α → E) (s : Set α) :
     ⨍ x in s, f x ∂μ = (μ s).toReal⁻¹ • ∫ x in s, f x ∂μ := by rw [average_eq, restrict_apply_univ]
 #align measure_theory.set_average_eq MeasureTheory.set_average_eq
+-/
 
+#print MeasureTheory.set_average_eq' /-
 theorem set_average_eq' (f : α → E) (s : Set α) :
     ⨍ x in s, f x ∂μ = ∫ x, f x ∂(μ s)⁻¹ • μ.restrict s := by
   simp only [average_eq', restrict_apply_univ]
 #align measure_theory.set_average_eq' MeasureTheory.set_average_eq'
+-/
 
 variable {μ}
 
+#print MeasureTheory.average_congr /-
 theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : ⨍ x, f x ∂μ = ⨍ x, g x ∂μ := by
   simp only [average_eq, integral_congr_ae h]
 #align measure_theory.average_congr MeasureTheory.average_congr
+-/
 
 theorem set_average_congr_set_ae (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ := by
   simp only [set_average_eq, set_integral_congr_set_ae h, measure_congr h]
 #align measure_theory.set_average_congr_set_ae MeasureTheory.set_average_congr_set_ae
 
+#print MeasureTheory.average_add_measure /-
 theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E}
     (hμ : Integrable f μ) (hν : Integrable f ν) :
     ⨍ x, f x ∂(μ + ν) =
@@ -162,17 +177,23 @@ theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeas
     integral_add_measure hμ hν, ← ENNReal.toReal_add (measure_ne_top μ _) (measure_ne_top ν _)]
   rw [average_eq, measure.add_apply]
 #align measure_theory.average_add_measure MeasureTheory.average_add_measure
+-/
 
+#print MeasureTheory.average_pair /-
 theorem average_pair {f : α → E} {g : α → F} (hfi : Integrable f μ) (hgi : Integrable g μ) :
     ⨍ x, (f x, g x) ∂μ = (⨍ x, f x ∂μ, ⨍ x, g x ∂μ) :=
   integral_pair hfi.to_average hgi.to_average
 #align measure_theory.average_pair MeasureTheory.average_pair
+-/
 
+#print MeasureTheory.measure_smul_set_average /-
 theorem measure_smul_set_average (f : α → E) {s : Set α} (h : μ s ≠ ∞) :
     (μ s).toReal • ⨍ x in s, f x ∂μ = ∫ x in s, f x ∂μ := by haveI := Fact.mk h.lt_top;
   rw [← measure_smul_average, restrict_apply_univ]
 #align measure_theory.measure_smul_set_average MeasureTheory.measure_smul_set_average
+-/
 
+#print MeasureTheory.average_union /-
 theorem average_union {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
     (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
     ⨍ x in s ∪ t, f x ∂μ =
@@ -182,7 +203,9 @@ theorem average_union {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t) (ht
   haveI := Fact.mk hsμ.lt_top; haveI := Fact.mk htμ.lt_top
   rw [restrict_union₀ hd ht, average_add_measure hfs hft, restrict_apply_univ, restrict_apply_univ]
 #align measure_theory.average_union MeasureTheory.average_union
+-/
 
+#print MeasureTheory.average_union_mem_openSegment /-
 theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t)
     (ht : NullMeasurableSet t μ) (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞)
     (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
@@ -194,7 +217,9 @@ theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AEDisj
     mem_open_segment_iff_div.mpr
       ⟨(μ s).toReal, (μ t).toReal, hs₀, ht₀, (average_union hd ht hsμ htμ hfs hft).symm⟩
 #align measure_theory.average_union_mem_open_segment MeasureTheory.average_union_mem_openSegment
+-/
 
+#print MeasureTheory.average_union_mem_segment /-
 theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t)
     (ht : NullMeasurableSet t μ) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ)
     (hft : IntegrableOn f t μ) : ⨍ x in s ∪ t, f x ∂μ ∈ [⨍ x in s, f x ∂μ -[ℝ] ⨍ x in t, f x ∂μ] :=
@@ -211,7 +236,9 @@ theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint
       0 < (μ s).toReal := ENNReal.toReal_pos hse hsμ
       _ ≤ _ := le_add_of_nonneg_right ENNReal.toReal_nonneg
 #align measure_theory.average_union_mem_segment MeasureTheory.average_union_mem_segment
+-/
 
+#print MeasureTheory.average_mem_openSegment_compl_self /-
 theorem average_mem_openSegment_compl_self [IsFiniteMeasure μ] {f : α → E} {s : Set α}
     (hs : NullMeasurableSet s μ) (hs₀ : μ s ≠ 0) (hsc₀ : μ (sᶜ) ≠ 0) (hfi : Integrable f μ) :
     ⨍ x, f x ∂μ ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in sᶜ, f x ∂μ) := by
@@ -219,7 +246,9 @@ theorem average_mem_openSegment_compl_self [IsFiniteMeasure μ] {f : α → E} {
     average_union_mem_open_segment ae_disjoint_compl_right hs.compl hs₀ hsc₀ (measure_ne_top _ _)
       (measure_ne_top _ _) hfi.integrable_on hfi.integrable_on
 #align measure_theory.average_mem_open_segment_compl_self MeasureTheory.average_mem_openSegment_compl_self
+-/
 
+#print MeasureTheory.average_const /-
 @[simp]
 theorem average_const [IsFiniteMeasure μ] [h : μ.ae.ne_bot] (c : E) : ⨍ x, c ∂μ = c := by
   simp only [average_eq, integral_const, measure.restrict_apply, MeasurableSet.univ, one_smul,
@@ -227,13 +256,16 @@ theorem average_const [IsFiniteMeasure μ] [h : μ.ae.ne_bot] (c : E) : ⨍ x, c
     measure_ne_top μ univ, Ne.def, measure_univ_eq_zero, ae_ne_bot.1 h, not_false_iff,
     ENNReal.one_toReal]
 #align measure_theory.average_const MeasureTheory.average_const
+-/
 
+#print MeasureTheory.set_average_const /-
 theorem set_average_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : E) : ⨍ x in s, c ∂μ = c :=
   by
   simp only [set_average_eq, integral_const, measure.restrict_apply, MeasurableSet.univ, univ_inter,
     smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel hs₀ hs,
     ENNReal.one_toReal, one_smul]
 #align measure_theory.set_average_const MeasureTheory.set_average_const
+-/
 
 @[simp]
 theorem integral_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :

bump to nightly-2023-06-12-03

mathlib commit https://github.com/leanprover-community/mathlib/commit/ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f

879d466d

Diff

@@ -1,10 +1,10 @@
 /-
 Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Yury G. Kudryashov
+Authors: Yury G. Kudryashov, Yaël Dillies
 
 ! This file was ported from Lean 3 source module measure_theory.integral.average
-! leanprover-community/mathlib commit c20927220ef87bb4962ba08bf6da2ce3cf50a6dd
+! leanprover-community/mathlib commit ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -24,25 +24,33 @@ measure, then the average of any function is equal to its integral.
 For the average on a set, we use `⨍ x in s, f x ∂μ` (notation for `⨍ x, f x ∂(μ.restrict s)`). For
 average w.r.t. the volume, one can omit `∂volume`.
 
+We prove several version of the first moment method: An integrable function is below/above its
+average on a set of positive measure.
+
 ## Implementation notes
 
 The average is defined as an integral over `(μ univ)⁻¹ • μ` so that all theorems about Bochner
 integrals work for the average without modifications. For theorems that require integrability of a
 function, we provide a convenience lemma `measure_theory.integrable.to_average`.
 
+## TODO
+
+Provide the first moment method for the Lebesgue integral as well. A draft is available on branch
+`first_moment_lintegral`.
+
 ## Tags
 
 integral, center mass, average value
 -/
 
 
-open MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
+open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
 
 open scoped Topology BigOperators ENNReal Convex
 
 variable {α E F : Type _} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E]
   [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {μ : Measure α}
-  {s : Set E}
+  {s t : Set α}
 
 /-!
 ### Average value of a function w.r.t. a measure
@@ -57,7 +65,9 @@ integral.
 
 namespace MeasureTheory
 
-variable (μ)
+section NormedAddCommGroup
+
+variable (μ) {f g : α → E}
 
 include m0
 
@@ -138,6 +148,10 @@ theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : ⨍ x, f x ∂μ = 
   simp only [average_eq, integral_congr_ae h]
 #align measure_theory.average_congr MeasureTheory.average_congr
 
+theorem set_average_congr_set_ae (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ := by
+  simp only [set_average_eq, set_integral_congr_set_ae h, measure_congr h]
+#align measure_theory.set_average_congr_set_ae MeasureTheory.set_average_congr_set_ae
+
 theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E}
     (hμ : Integrable f μ) (hν : Integrable f ν) :
     ⨍ x, f x ∂(μ + ν) =
@@ -221,5 +235,238 @@ theorem set_average_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞)
     ENNReal.one_toReal, one_smul]
 #align measure_theory.set_average_const MeasureTheory.set_average_const
 
+@[simp]
+theorem integral_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :
+    ∫ x, ⨍ a, f a ∂μ ∂μ = ∫ x, f x ∂μ :=
+  by
+  obtain rfl | hμ := eq_or_ne μ 0
+  · simp only [integral_zero_measure]
+  ·
+    rw [integral_const, average_eq,
+      smul_inv_smul₀ (to_real_ne_zero.2 ⟨measure_univ_ne_zero.2 hμ, measure_ne_top _ _⟩)]
+#align measure_theory.integral_average MeasureTheory.integral_average
+
+theorem set_integral_set_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) (s : Set α) :
+    ∫ x in s, ⨍ a in s, f a ∂μ ∂μ = ∫ x in s, f x ∂μ :=
+  integral_average _ _
+#align measure_theory.set_integral_set_average MeasureTheory.set_integral_set_average
+
+theorem integral_sub_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :
+    ∫ x, f x - ⨍ a, f a ∂μ ∂μ = 0 :=
+  by
+  by_cases hf : integrable f μ
+  · rw [integral_sub hf (integrable_const _), integral_average, sub_self]
+  refine' integral_undef fun h => hf _
+  convert h.add (integrable_const _)
+  exact (sub_add_cancel _ _).symm
+#align measure_theory.integral_sub_average MeasureTheory.integral_sub_average
+
+theorem set_integral_sub_set_average (hs : μ s ≠ ∞) (f : α → E) :
+    ∫ x in s, f x - ⨍ a in s, f a ∂μ ∂μ = 0 :=
+  haveI haveI : Fact (μ s < ∞) := ⟨lt_top_iff_ne_top.2 hs⟩
+  integral_sub_average _ _
+#align measure_theory.set_integral_sub_set_average MeasureTheory.set_integral_sub_set_average
+
+theorem integral_average_sub [IsFiniteMeasure μ] (hf : Integrable f μ) :
+    ∫ x, ⨍ a, f a ∂μ - f x ∂μ = 0 := by
+  rw [integral_sub (integrable_const _) hf, integral_average, sub_self]
+#align measure_theory.integral_average_sub MeasureTheory.integral_average_sub
+
+theorem set_integral_set_average_sub (hs : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+    ∫ x in s, ⨍ a in s, f a ∂μ - f x ∂μ = 0 :=
+  haveI haveI : Fact (μ s < ∞) := ⟨lt_top_iff_ne_top.2 hs⟩
+  integral_average_sub hf
+#align measure_theory.set_integral_set_average_sub MeasureTheory.set_integral_set_average_sub
+
+end NormedAddCommGroup
+
+theorem ofReal_average {f : α → ℝ} (hf : Integrable f μ) (hf₀ : 0 ≤ᵐ[μ] f) :
+    ENNReal.ofReal (⨍ x, f x ∂μ) = (∫⁻ x, ENNReal.ofReal (f x) ∂μ) / μ univ :=
+  by
+  obtain rfl | hμ := eq_or_ne μ 0
+  · simp
+  ·
+    rw [average_eq, smul_eq_mul, ← to_real_inv, of_real_mul to_real_nonneg,
+      of_real_to_real (inv_ne_top.2 <| measure_univ_ne_zero.2 hμ),
+      of_real_integral_eq_lintegral_of_real hf hf₀, ENNReal.div_eq_inv_mul]
+#align measure_theory.of_real_average MeasureTheory.ofReal_average
+
+theorem ofReal_set_average {f : α → ℝ} (hf : IntegrableOn f s μ) (hf₀ : 0 ≤ᵐ[μ.restrict s] f) :
+    ENNReal.ofReal (⨍ x in s, f x ∂μ) = (∫⁻ x in s, ENNReal.ofReal (f x) ∂μ) / μ s := by
+  simpa using of_real_average hf hf₀
+#align measure_theory.of_real_set_average MeasureTheory.ofReal_set_average
+
+theorem average_toReal {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf' : ∀ᵐ x ∂μ, f x ≠ ∞) :
+    ⨍ x, (f x).toReal ∂μ = ((∫⁻ x, f x ∂μ) / μ univ).toReal :=
+  by
+  obtain rfl | hμ := eq_or_ne μ 0
+  · simp
+  ·
+    rw [average_eq, smul_eq_mul, to_real_div, ←
+      integral_to_real hf (hf'.mono fun _ => lt_top_iff_ne_top.2), div_eq_inv_mul]
+#align measure_theory.average_to_real MeasureTheory.average_toReal
+
+theorem set_average_toReal {f : α → ℝ≥0∞} (hf : AEMeasurable f (μ.restrict s))
+    (hf' : ∀ᵐ x ∂μ.restrict s, f x ≠ ∞) :
+    ⨍ x in s, (f x).toReal ∂μ = ((∫⁻ x in s, f x ∂μ) / μ s).toReal := by
+  simpa using average_to_real hf hf'
+#align measure_theory.set_average_to_real MeasureTheory.set_average_toReal
+
+/-! ### First moment method -/
+
+
+section FirstMoment
+
+variable {N : Set α} {f : α → ℝ}
+
+/-- **First moment method**. An integrable function is smaller than its mean on a set of positive
+measure. -/
+theorem measure_le_set_average_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+    0 < μ ({x ∈ s | f x ≤ ⨍ a in s, f a ∂μ}) :=
+  by
+  refine' pos_iff_ne_zero.2 fun H => _
+  replace H : (μ.restrict s) {x | f x ≤ ⨍ a in s, f a ∂μ} = 0
+  · rwa [restrict_apply₀, inter_comm]
+    exact ae_strongly_measurable.null_measurable_set_le hf.1 ae_strongly_measurable_const
+  haveI : is_finite_measure (μ.restrict s) :=
+    ⟨by simpa only [measure.restrict_apply, MeasurableSet.univ, univ_inter] using hμ₁.lt_top⟩
+  refine' (integral_sub_average (μ.restrict s) f).not_gt _
+  refine' (set_integral_pos_iff_support_of_nonneg_ae _ _).2 _
+  · refine' eq_bot_mono (measure_mono fun x hx => _) H
+    simp only [Pi.zero_apply, sub_nonneg, mem_compl_iff, mem_set_of_eq, not_le] at hx 
+    exact hx.le
+  · exact hf.sub (integrable_on_const.2 <| Or.inr <| lt_top_iff_ne_top.2 hμ₁)
+  · rwa [pos_iff_ne_zero, inter_comm, ← diff_compl, ← diff_inter_self_eq_diff, measure_diff_null]
+    refine' eq_bot_mono (measure_mono _) (measure_inter_eq_zero_of_restrict H)
+    exact inter_subset_inter_left _ fun a ha => (sub_eq_zero.1 <| of_not_not ha).le
+#align measure_theory.measure_le_set_average_pos MeasureTheory.measure_le_set_average_pos
+
+/-- **First moment method**. An integrable function is greater than its mean on a set of positive
+measure. -/
+theorem measure_set_average_le_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+    0 < μ ({x ∈ s | ⨍ a in s, f a ∂μ ≤ f x}) := by
+  simpa [integral_neg, neg_div] using measure_le_set_average_pos hμ hμ₁ hf.neg
+#align measure_theory.measure_set_average_le_pos MeasureTheory.measure_set_average_le_pos
+
+/-- **First moment method**. The minimum of an integrable function is smaller than its mean. -/
+theorem exists_le_set_average (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+    ∃ x ∈ s, f x ≤ ⨍ a in s, f a ∂μ :=
+  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_le_set_average_pos hμ hμ₁ hf).ne'
+  ⟨x, hx, h⟩
+#align measure_theory.exists_le_set_average MeasureTheory.exists_le_set_average
+
+/-- **First moment method**. The maximum of an integrable function is greater than its mean. -/
+theorem exists_set_average_le (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+    ∃ x ∈ s, ⨍ a in s, f a ∂μ ≤ f x :=
+  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_set_average_le_pos hμ hμ₁ hf).ne'
+  ⟨x, hx, h⟩
+#align measure_theory.exists_set_average_le MeasureTheory.exists_set_average_le
+
+section FiniteMeasure
+
+variable [IsFiniteMeasure μ]
+
+/-- **First moment method**. An integrable function is smaller than its mean on a set of positive
+measure. -/
+theorem measure_le_average_pos (hμ : μ ≠ 0) (hf : Integrable f μ) : 0 < μ {x | f x ≤ ⨍ a, f a ∂μ} :=
+  by
+  simpa using
+    measure_le_set_average_pos (measure.measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)
+      hf.integrable_on
+#align measure_theory.measure_le_average_pos MeasureTheory.measure_le_average_pos
+
+/-- **First moment method**. An integrable function is greater than its mean on a set of positive
+measure. -/
+theorem measure_average_le_pos (hμ : μ ≠ 0) (hf : Integrable f μ) : 0 < μ {x | ⨍ a, f a ∂μ ≤ f x} :=
+  by
+  simpa using
+    measure_set_average_le_pos (measure.measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)
+      hf.integrable_on
+#align measure_theory.measure_average_le_pos MeasureTheory.measure_average_le_pos
+
+/-- **First moment method**. The minimum of an integrable function is smaller than its mean. -/
+theorem exists_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) : ∃ x, f x ≤ ⨍ a, f a ∂μ :=
+  let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_le_average_pos hμ hf).ne'
+  ⟨x, hx⟩
+#align measure_theory.exists_le_average MeasureTheory.exists_le_average
+
+/-- **First moment method**. The maximum of an integrable function is greater than its mean. -/
+theorem exists_average_le (hμ : μ ≠ 0) (hf : Integrable f μ) : ∃ x, ⨍ a, f a ∂μ ≤ f x :=
+  let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_average_le_pos hμ hf).ne'
+  ⟨x, hx⟩
+#align measure_theory.exists_average_le MeasureTheory.exists_average_le
+
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/-- **First moment method**. The minimum of an integrable function is smaller than its mean, while
+avoiding a null set. -/
+theorem exists_not_mem_null_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) (hN : μ N = 0) :
+    ∃ (x : _) (_ : x ∉ N), f x ≤ ⨍ a, f a ∂μ :=
+  by
+  have := measure_le_average_pos hμ hf
+  rw [← measure_diff_null hN] at this 
+  obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
+  exact ⟨x, hxN, hx⟩
+#align measure_theory.exists_not_mem_null_le_average MeasureTheory.exists_not_mem_null_le_average
+
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/-- **First moment method**. The maximum of an integrable function is greater than its mean, while
+avoiding a null set. -/
+theorem exists_not_mem_null_average_le (hμ : μ ≠ 0) (hf : Integrable f μ) (hN : μ N = 0) :
+    ∃ (x : _) (_ : x ∉ N), ⨍ a, f a ∂μ ≤ f x := by
+  simpa [integral_neg, neg_div] using exists_not_mem_null_le_average hμ hf.neg hN
+#align measure_theory.exists_not_mem_null_average_le MeasureTheory.exists_not_mem_null_average_le
+
+end FiniteMeasure
+
+section ProbabilityMeasure
+
+variable [IsProbabilityMeasure μ]
+
+/-- **First moment method**. An integrable function is smaller than its integral on a set of
+positive measure. -/
+theorem measure_le_integral_pos (hf : Integrable f μ) : 0 < μ {x | f x ≤ ∫ a, f a ∂μ} := by
+  simpa only [average_eq_integral] using
+    measure_le_average_pos (is_probability_measure.ne_zero μ) hf
+#align measure_theory.measure_le_integral_pos MeasureTheory.measure_le_integral_pos
+
+/-- **First moment method**. An integrable function is greater than its integral on a set of
+positive measure. -/
+theorem measure_integral_le_pos (hf : Integrable f μ) : 0 < μ {x | ∫ a, f a ∂μ ≤ f x} := by
+  simpa only [average_eq_integral] using
+    measure_average_le_pos (is_probability_measure.ne_zero μ) hf
+#align measure_theory.measure_integral_le_pos MeasureTheory.measure_integral_le_pos
+
+/-- **First moment method**. The minimum of an integrable function is smaller than its integral. -/
+theorem exists_le_integral (hf : Integrable f μ) : ∃ x, f x ≤ ∫ a, f a ∂μ := by
+  simpa only [average_eq_integral] using exists_le_average (is_probability_measure.ne_zero μ) hf
+#align measure_theory.exists_le_integral MeasureTheory.exists_le_integral
+
+/-- **First moment method**. The maximum of an integrable function is greater than its integral. -/
+theorem exists_integral_le (hf : Integrable f μ) : ∃ x, ∫ a, f a ∂μ ≤ f x := by
+  simpa only [average_eq_integral] using exists_average_le (is_probability_measure.ne_zero μ) hf
+#align measure_theory.exists_integral_le MeasureTheory.exists_integral_le
+
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/-- **First moment method**. The minimum of an integrable function is smaller than its integral,
+while avoiding a null set. -/
+theorem exists_not_mem_null_le_integral (hf : Integrable f μ) (hN : μ N = 0) :
+    ∃ (x : _) (_ : x ∉ N), f x ≤ ∫ a, f a ∂μ := by
+  simpa only [average_eq_integral] using
+    exists_not_mem_null_le_average (is_probability_measure.ne_zero μ) hf hN
+#align measure_theory.exists_not_mem_null_le_integral MeasureTheory.exists_not_mem_null_le_integral
+
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » N) -/
+/-- **First moment method**. The maximum of an integrable function is greater than its integral,
+while avoiding a null set. -/
+theorem exists_not_mem_null_integral_le (hf : Integrable f μ) (hN : μ N = 0) :
+    ∃ (x : _) (_ : x ∉ N), ∫ a, f a ∂μ ≤ f x := by
+  simpa only [average_eq_integral] using
+    exists_not_mem_null_average_le (is_probability_measure.ne_zero μ) hf hN
+#align measure_theory.exists_not_mem_null_integral_le MeasureTheory.exists_not_mem_null_integral_le
+
+end ProbabilityMeasure
+
+end FirstMoment
+
 end MeasureTheory

c2092722

bump to nightly-2023-06-10-07

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

3fdc57bc

Diff

@@ -75,46 +75,46 @@ noncomputable def average (f : α → E) :=
 -/
 
 -- mathport name: «expr⨍ , ∂ »
-notation3"⨍ "(...)", "r:(scoped f => f)" ∂"μ => average μ r
+notation3"⨍ "(...)", "r:60:(scoped f => f)" ∂"μ:70 => average μ r
 
 -- mathport name: «expr⨍ , »
-notation3"⨍ "(...)", "r:(scoped f => average volume f) => r
+notation3"⨍ "(...)", "r:60:(scoped f => average volume f) => r
 
 -- mathport name: «expr⨍ in , ∂ »
-notation3"⨍ "(...)" in "s", "r:(scoped f => f)" ∂"μ => average (Measure.restrict μ s) r
+notation3"⨍ "(...)" in "s", "r:60:(scoped f => f)" ∂"μ:70 => average (Measure.restrict μ s) r
 
 -- mathport name: «expr⨍ in , »
-notation3"⨍ "(...)" in "s", "r:(scoped f => average Measure.restrict volume s f) => r
+notation3"⨍ "(...)" in "s", "r:60:(scoped f => average Measure.restrict volume s f) => r
 
 @[simp]
-theorem average_zero : (⨍ x, (0 : E) ∂μ) = 0 := by rw [average, integral_zero]
+theorem average_zero : ⨍ x, (0 : E) ∂μ = 0 := by rw [average, integral_zero]
 #align measure_theory.average_zero MeasureTheory.average_zero
 
 @[simp]
-theorem average_zero_measure (f : α → E) : (⨍ x, f x ∂(0 : Measure α)) = 0 := by
+theorem average_zero_measure (f : α → E) : ⨍ x, f x ∂(0 : Measure α) = 0 := by
   rw [average, smul_zero, integral_zero_measure]
 #align measure_theory.average_zero_measure MeasureTheory.average_zero_measure
 
 @[simp]
-theorem average_neg (f : α → E) : (⨍ x, -f x ∂μ) = -⨍ x, f x ∂μ :=
+theorem average_neg (f : α → E) : ⨍ x, -f x ∂μ = -⨍ x, f x ∂μ :=
   integral_neg f
 #align measure_theory.average_neg MeasureTheory.average_neg
 
-theorem average_eq' (f : α → E) : (⨍ x, f x ∂μ) = ∫ x, f x ∂(μ univ)⁻¹ • μ :=
+theorem average_eq' (f : α → E) : ⨍ x, f x ∂μ = ∫ x, f x ∂(μ univ)⁻¹ • μ :=
   rfl
 #align measure_theory.average_eq' MeasureTheory.average_eq'
 
-theorem average_eq (f : α → E) : (⨍ x, f x ∂μ) = (μ univ).toReal⁻¹ • ∫ x, f x ∂μ := by
+theorem average_eq (f : α → E) : ⨍ x, f x ∂μ = (μ univ).toReal⁻¹ • ∫ x, f x ∂μ := by
   rw [average_eq', integral_smul_measure, ENNReal.toReal_inv]
 #align measure_theory.average_eq MeasureTheory.average_eq
 
-theorem average_eq_integral [IsProbabilityMeasure μ] (f : α → E) : (⨍ x, f x ∂μ) = ∫ x, f x ∂μ := by
+theorem average_eq_integral [IsProbabilityMeasure μ] (f : α → E) : ⨍ x, f x ∂μ = ∫ x, f x ∂μ := by
   rw [average, measure_univ, inv_one, one_smul]
 #align measure_theory.average_eq_integral MeasureTheory.average_eq_integral
 
 @[simp]
 theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
-    ((μ univ).toReal • ⨍ x, f x ∂μ) = ∫ x, f x ∂μ :=
+    (μ univ).toReal • ⨍ x, f x ∂μ = ∫ x, f x ∂μ :=
   by
   cases' eq_or_ne μ 0 with hμ hμ
   · rw [hμ, integral_zero_measure, average_zero_measure, smul_zero]
@@ -124,25 +124,24 @@ theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
 #align measure_theory.measure_smul_average MeasureTheory.measure_smul_average
 
 theorem set_average_eq (f : α → E) (s : Set α) :
-    (⨍ x in s, f x ∂μ) = (μ s).toReal⁻¹ • ∫ x in s, f x ∂μ := by
-  rw [average_eq, restrict_apply_univ]
+    ⨍ x in s, f x ∂μ = (μ s).toReal⁻¹ • ∫ x in s, f x ∂μ := by rw [average_eq, restrict_apply_univ]
 #align measure_theory.set_average_eq MeasureTheory.set_average_eq
 
 theorem set_average_eq' (f : α → E) (s : Set α) :
-    (⨍ x in s, f x ∂μ) = ∫ x, f x ∂(μ s)⁻¹ • μ.restrict s := by
+    ⨍ x in s, f x ∂μ = ∫ x, f x ∂(μ s)⁻¹ • μ.restrict s := by
   simp only [average_eq', restrict_apply_univ]
 #align measure_theory.set_average_eq' MeasureTheory.set_average_eq'
 
 variable {μ}
 
-theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : (⨍ x, f x ∂μ) = ⨍ x, g x ∂μ := by
+theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : ⨍ x, f x ∂μ = ⨍ x, g x ∂μ := by
   simp only [average_eq, integral_congr_ae h]
 #align measure_theory.average_congr MeasureTheory.average_congr
 
 theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E}
     (hμ : Integrable f μ) (hν : Integrable f ν) :
-    (⨍ x, f x ∂μ + ν) =
-      (((μ univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂μ) +
+    ⨍ x, f x ∂(μ + ν) =
+      ((μ univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂μ +
         ((ν univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂ν :=
   by
   simp only [div_eq_inv_mul, mul_smul, measure_smul_average, ← smul_add, ←
@@ -151,19 +150,19 @@ theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeas
 #align measure_theory.average_add_measure MeasureTheory.average_add_measure
 
 theorem average_pair {f : α → E} {g : α → F} (hfi : Integrable f μ) (hgi : Integrable g μ) :
-    (⨍ x, (f x, g x) ∂μ) = (⨍ x, f x ∂μ, ⨍ x, g x ∂μ) :=
+    ⨍ x, (f x, g x) ∂μ = (⨍ x, f x ∂μ, ⨍ x, g x ∂μ) :=
   integral_pair hfi.to_average hgi.to_average
 #align measure_theory.average_pair MeasureTheory.average_pair
 
 theorem measure_smul_set_average (f : α → E) {s : Set α} (h : μ s ≠ ∞) :
-    ((μ s).toReal • ⨍ x in s, f x ∂μ) = ∫ x in s, f x ∂μ := by haveI := Fact.mk h.lt_top;
+    (μ s).toReal • ⨍ x in s, f x ∂μ = ∫ x in s, f x ∂μ := by haveI := Fact.mk h.lt_top;
   rw [← measure_smul_average, restrict_apply_univ]
 #align measure_theory.measure_smul_set_average MeasureTheory.measure_smul_set_average
 
 theorem average_union {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
     (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
-    (⨍ x in s ∪ t, f x ∂μ) =
-      (((μ s).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ x in s, f x ∂μ) +
+    ⨍ x in s ∪ t, f x ∂μ =
+      ((μ s).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ x in s, f x ∂μ +
         ((μ t).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ x in t, f x ∂μ :=
   by
   haveI := Fact.mk hsμ.lt_top; haveI := Fact.mk htμ.lt_top
@@ -173,7 +172,7 @@ theorem average_union {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t) (ht
 theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t)
     (ht : NullMeasurableSet t μ) (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞)
     (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
-    (⨍ x in s ∪ t, f x ∂μ) ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in t, f x ∂μ) :=
+    ⨍ x in s ∪ t, f x ∂μ ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in t, f x ∂μ) :=
   by
   replace hs₀ : 0 < (μ s).toReal; exact ENNReal.toReal_pos hs₀ hsμ
   replace ht₀ : 0 < (μ t).toReal; exact ENNReal.toReal_pos ht₀ htμ
@@ -184,8 +183,7 @@ theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AEDisj
 
 theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t)
     (ht : NullMeasurableSet t μ) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ)
-    (hft : IntegrableOn f t μ) :
-    (⨍ x in s ∪ t, f x ∂μ) ∈ [⨍ x in s, f x ∂μ -[ℝ] ⨍ x in t, f x ∂μ] :=
+    (hft : IntegrableOn f t μ) : ⨍ x in s ∪ t, f x ∂μ ∈ [⨍ x in s, f x ∂μ -[ℝ] ⨍ x in t, f x ∂μ] :=
   by
   by_cases hse : μ s = 0
   · rw [← ae_eq_empty] at hse 
@@ -202,22 +200,22 @@ theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint
 
 theorem average_mem_openSegment_compl_self [IsFiniteMeasure μ] {f : α → E} {s : Set α}
     (hs : NullMeasurableSet s μ) (hs₀ : μ s ≠ 0) (hsc₀ : μ (sᶜ) ≠ 0) (hfi : Integrable f μ) :
-    (⨍ x, f x ∂μ) ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in sᶜ, f x ∂μ) := by
+    ⨍ x, f x ∂μ ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in sᶜ, f x ∂μ) := by
   simpa only [union_compl_self, restrict_univ] using
     average_union_mem_open_segment ae_disjoint_compl_right hs.compl hs₀ hsc₀ (measure_ne_top _ _)
       (measure_ne_top _ _) hfi.integrable_on hfi.integrable_on
 #align measure_theory.average_mem_open_segment_compl_self MeasureTheory.average_mem_openSegment_compl_self
 
 @[simp]
-theorem average_const [IsFiniteMeasure μ] [h : μ.ae.ne_bot] (c : E) : (⨍ x, c ∂μ) = c := by
+theorem average_const [IsFiniteMeasure μ] [h : μ.ae.ne_bot] (c : E) : ⨍ x, c ∂μ = c := by
   simp only [average_eq, integral_const, measure.restrict_apply, MeasurableSet.univ, one_smul,
     univ_inter, smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel,
     measure_ne_top μ univ, Ne.def, measure_univ_eq_zero, ae_ne_bot.1 h, not_false_iff,
     ENNReal.one_toReal]
 #align measure_theory.average_const MeasureTheory.average_const
 
-theorem set_average_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : E) :
-    (⨍ x in s, c ∂μ) = c := by
+theorem set_average_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : E) : ⨍ x in s, c ∂μ = c :=
+  by
   simp only [set_average_eq, integral_const, measure.restrict_apply, MeasurableSet.univ, univ_inter,
     smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel hs₀ hs,
     ENNReal.one_toReal, one_smul]

c2092722

bump to nightly-2023-06-09-07

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

16afdc39

Diff

@@ -198,7 +198,6 @@ theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint
     calc
       0 < (μ s).toReal := ENNReal.toReal_pos hse hsμ
       _ ≤ _ := le_add_of_nonneg_right ENNReal.toReal_nonneg
-      
 #align measure_theory.average_union_mem_segment MeasureTheory.average_union_mem_segment
 
 theorem average_mem_openSegment_compl_self [IsFiniteMeasure μ] {f : α → E} {s : Set α}

c2092722

bump to nightly-2023-06-08-03

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

8f04dc2b

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov
 
 ! This file was ported from Lean 3 source module measure_theory.integral.average
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit c20927220ef87bb4962ba08bf6da2ce3cf50a6dd
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.MeasureTheory.Integral.SetIntegral
 /-!
 # Integral average of a function
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we define `measure_theory.average μ f` (notation: `⨍ x, f x ∂μ`) to be the average
 value of `f` with respect to measure `μ`. It is defined as `∫ x, f x ∂((μ univ)⁻¹ • μ)`, so it
 is equal to zero if `f` is not integrable or if `μ` is an infinite measure. If `μ` is a probability

bump to nightly-2023-06-06-07

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3209ddf94136d36e5e5c624b10b2a347cc9d090

152a172d

Diff

@@ -58,6 +58,7 @@ variable (μ)
 
 include m0
 
+#print MeasureTheory.average /-
 /-- Average value of a function `f` w.r.t. a measure `μ`, notation: `⨍ x, f x ∂μ`. It is defined as
 `(μ univ).to_real⁻¹ • ∫ x, f x ∂μ`, so it is equal to zero if `f` is not integrable or if `μ` is an
 infinite measure. If `μ` is a probability measure, then the average of any function is equal to its
@@ -68,6 +69,7 @@ average w.r.t. the volume, one can omit `∂volume`. -/
 noncomputable def average (f : α → E) :=
   ∫ x, f x ∂(μ univ)⁻¹ • μ
 #align measure_theory.average MeasureTheory.average
+-/
 
 -- mathport name: «expr⨍ , ∂ »
 notation3"⨍ "(...)", "r:(scoped f => f)" ∂"μ => average μ r

bump to nightly-2023-06-04-18

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

3fb4268b

Diff

@@ -103,12 +103,12 @@ theorem average_eq (f : α → E) : (⨍ x, f x ∂μ) = (μ univ).toReal⁻¹ 
   rw [average_eq', integral_smul_measure, ENNReal.toReal_inv]
 #align measure_theory.average_eq MeasureTheory.average_eq
 
-theorem average_eq_integral [ProbabilityMeasure μ] (f : α → E) : (⨍ x, f x ∂μ) = ∫ x, f x ∂μ := by
+theorem average_eq_integral [IsProbabilityMeasure μ] (f : α → E) : (⨍ x, f x ∂μ) = ∫ x, f x ∂μ := by
   rw [average, measure_univ, inv_one, one_smul]
 #align measure_theory.average_eq_integral MeasureTheory.average_eq_integral
 
 @[simp]
-theorem measure_smul_average [FiniteMeasure μ] (f : α → E) :
+theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
     ((μ univ).toReal • ⨍ x, f x ∂μ) = ∫ x, f x ∂μ :=
   by
   cases' eq_or_ne μ 0 with hμ hμ
@@ -134,7 +134,7 @@ theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : (⨍ x, f x ∂μ) =
   simp only [average_eq, integral_congr_ae h]
 #align measure_theory.average_congr MeasureTheory.average_congr
 
-theorem average_add_measure [FiniteMeasure μ] {ν : Measure α} [FiniteMeasure ν] {f : α → E}
+theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E}
     (hμ : Integrable f μ) (hν : Integrable f ν) :
     (⨍ x, f x ∂μ + ν) =
       (((μ univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂μ) +
@@ -196,7 +196,7 @@ theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint
       
 #align measure_theory.average_union_mem_segment MeasureTheory.average_union_mem_segment
 
-theorem average_mem_openSegment_compl_self [FiniteMeasure μ] {f : α → E} {s : Set α}
+theorem average_mem_openSegment_compl_self [IsFiniteMeasure μ] {f : α → E} {s : Set α}
     (hs : NullMeasurableSet s μ) (hs₀ : μ s ≠ 0) (hsc₀ : μ (sᶜ) ≠ 0) (hfi : Integrable f μ) :
     (⨍ x, f x ∂μ) ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in sᶜ, f x ∂μ) := by
   simpa only [union_compl_self, restrict_univ] using
@@ -205,7 +205,7 @@ theorem average_mem_openSegment_compl_self [FiniteMeasure μ] {f : α → E} {s
 #align measure_theory.average_mem_open_segment_compl_self MeasureTheory.average_mem_openSegment_compl_self
 
 @[simp]
-theorem average_const [FiniteMeasure μ] [h : μ.ae.ne_bot] (c : E) : (⨍ x, c ∂μ) = c := by
+theorem average_const [IsFiniteMeasure μ] [h : μ.ae.ne_bot] (c : E) : (⨍ x, c ∂μ) = c := by
   simp only [average_eq, integral_const, measure.restrict_apply, MeasurableSet.univ, one_smul,
     univ_inter, smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel,
     measure_ne_top μ univ, Ne.def, measure_univ_eq_zero, ae_ne_bot.1 h, not_false_iff,

bump to nightly-2023-06-01-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

b9c87c70

Diff

@@ -183,7 +183,7 @@ theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint
     (⨍ x in s ∪ t, f x ∂μ) ∈ [⨍ x in s, f x ∂μ -[ℝ] ⨍ x in t, f x ∂μ] :=
   by
   by_cases hse : μ s = 0
-  · rw [← ae_eq_empty] at hse
+  · rw [← ae_eq_empty] at hse 
     rw [restrict_congr_set (hse.union eventually_eq.rfl), empty_union]
     exact right_mem_segment _ _ _
   · refine'

bump to nightly-2023-05-28-18

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

8d353152

Diff

@@ -35,7 +35,7 @@ integral, center mass, average value
 
 open MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
 
-open Topology BigOperators ENNReal Convex
+open scoped Topology BigOperators ENNReal Convex
 
 variable {α E F : Type _} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E]
   [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {μ : Measure α}

bump to nightly-2023-05-28-04

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

6797a637

Diff

@@ -151,9 +151,7 @@ theorem average_pair {f : α → E} {g : α → F} (hfi : Integrable f μ) (hgi
 #align measure_theory.average_pair MeasureTheory.average_pair
 
 theorem measure_smul_set_average (f : α → E) {s : Set α} (h : μ s ≠ ∞) :
-    ((μ s).toReal • ⨍ x in s, f x ∂μ) = ∫ x in s, f x ∂μ :=
-  by
-  haveI := Fact.mk h.lt_top
+    ((μ s).toReal • ⨍ x in s, f x ∂μ) = ∫ x in s, f x ∂μ := by haveI := Fact.mk h.lt_top;
   rw [← measure_smul_average, restrict_apply_univ]
 #align measure_theory.measure_smul_set_average MeasureTheory.measure_smul_set_average

bump to nightly-2023-05-09-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/d4437c68c8d350fc9d4e95e1e174409db35e30d7

244524af

Diff

@@ -103,12 +103,12 @@ theorem average_eq (f : α → E) : (⨍ x, f x ∂μ) = (μ univ).toReal⁻¹ 
   rw [average_eq', integral_smul_measure, ENNReal.toReal_inv]
 #align measure_theory.average_eq MeasureTheory.average_eq
 
-theorem average_eq_integral [IsProbabilityMeasure μ] (f : α → E) : (⨍ x, f x ∂μ) = ∫ x, f x ∂μ := by
+theorem average_eq_integral [ProbabilityMeasure μ] (f : α → E) : (⨍ x, f x ∂μ) = ∫ x, f x ∂μ := by
   rw [average, measure_univ, inv_one, one_smul]
 #align measure_theory.average_eq_integral MeasureTheory.average_eq_integral
 
 @[simp]
-theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
+theorem measure_smul_average [FiniteMeasure μ] (f : α → E) :
     ((μ univ).toReal • ⨍ x, f x ∂μ) = ∫ x, f x ∂μ :=
   by
   cases' eq_or_ne μ 0 with hμ hμ
@@ -134,7 +134,7 @@ theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : (⨍ x, f x ∂μ) =
   simp only [average_eq, integral_congr_ae h]
 #align measure_theory.average_congr MeasureTheory.average_congr
 
-theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E}
+theorem average_add_measure [FiniteMeasure μ] {ν : Measure α} [FiniteMeasure ν] {f : α → E}
     (hμ : Integrable f μ) (hν : Integrable f ν) :
     (⨍ x, f x ∂μ + ν) =
       (((μ univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂μ) +
@@ -198,7 +198,7 @@ theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint
       
 #align measure_theory.average_union_mem_segment MeasureTheory.average_union_mem_segment
 
-theorem average_mem_openSegment_compl_self [IsFiniteMeasure μ] {f : α → E} {s : Set α}
+theorem average_mem_openSegment_compl_self [FiniteMeasure μ] {f : α → E} {s : Set α}
     (hs : NullMeasurableSet s μ) (hs₀ : μ s ≠ 0) (hsc₀ : μ (sᶜ) ≠ 0) (hfi : Integrable f μ) :
     (⨍ x, f x ∂μ) ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in sᶜ, f x ∂μ) := by
   simpa only [union_compl_self, restrict_univ] using
@@ -207,7 +207,7 @@ theorem average_mem_openSegment_compl_self [IsFiniteMeasure μ] {f : α → E} {
 #align measure_theory.average_mem_open_segment_compl_self MeasureTheory.average_mem_openSegment_compl_self
 
 @[simp]
-theorem average_const [IsFiniteMeasure μ] [h : μ.ae.ne_bot] (c : E) : (⨍ x, c ∂μ) = c := by
+theorem average_const [FiniteMeasure μ] [h : μ.ae.ne_bot] (c : E) : (⨍ x, c ∂μ) = c := by
   simp only [average_eq, integral_const, measure.restrict_apply, MeasurableSet.univ, one_smul,
     univ_inter, smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel,
     measure_ne_top μ univ, Ne.def, measure_univ_eq_zero, ae_ne_bot.1 h, not_false_iff,

bump to nightly-2023-05-06-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/3905fa80e62c0898131285baab35559fbc4e5cda

e8904d96

Diff

@@ -157,7 +157,7 @@ theorem measure_smul_set_average (f : α → E) {s : Set α} (h : μ s ≠ ∞)
   rw [← measure_smul_average, restrict_apply_univ]
 #align measure_theory.measure_smul_set_average MeasureTheory.measure_smul_set_average
 
-theorem average_union {f : α → E} {s t : Set α} (hd : AeDisjoint μ s t) (ht : NullMeasurableSet t μ)
+theorem average_union {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
     (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
     (⨍ x in s ∪ t, f x ∂μ) =
       (((μ s).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ x in s, f x ∂μ) +
@@ -167,7 +167,7 @@ theorem average_union {f : α → E} {s t : Set α} (hd : AeDisjoint μ s t) (ht
   rw [restrict_union₀ hd ht, average_add_measure hfs hft, restrict_apply_univ, restrict_apply_univ]
 #align measure_theory.average_union MeasureTheory.average_union
 
-theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AeDisjoint μ s t)
+theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t)
     (ht : NullMeasurableSet t μ) (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞)
     (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
     (⨍ x in s ∪ t, f x ∂μ) ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in t, f x ∂μ) :=
@@ -179,7 +179,7 @@ theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AeDisj
       ⟨(μ s).toReal, (μ t).toReal, hs₀, ht₀, (average_union hd ht hsμ htμ hfs hft).symm⟩
 #align measure_theory.average_union_mem_open_segment MeasureTheory.average_union_mem_openSegment
 
-theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AeDisjoint μ s t)
+theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t)
     (ht : NullMeasurableSet t μ) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ)
     (hft : IntegrableOn f t μ) :
     (⨍ x in s ∪ t, f x ∂μ) ∈ [⨍ x in s, f x ∂μ -[ℝ] ⨍ x in t, f x ∂μ] :=

bump to nightly-2023-05-04-10

mathlib commit https://github.com/leanprover-community/mathlib/commit/92c69b77c5a7dc0f7eeddb552508633305157caa

3d8894bd

Diff

@@ -147,7 +147,7 @@ theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeas
 
 theorem average_pair {f : α → E} {g : α → F} (hfi : Integrable f μ) (hgi : Integrable g μ) :
     (⨍ x, (f x, g x) ∂μ) = (⨍ x, f x ∂μ, ⨍ x, g x ∂μ) :=
-  integral_pair hfi.toAverage hgi.toAverage
+  integral_pair hfi.to_average hgi.to_average
 #align measure_theory.average_pair MeasureTheory.average_pair
 
 theorem measure_smul_set_average (f : α → E) {s : Set α} (h : μ s ≠ ∞) :

bump to nightly-2023-02-27-10

mathlib commit https://github.com/leanprover-community/mathlib/commit/eb0cb4511aaef0da2462207b67358a0e1fe1e2ee

49ff026a

Diff

@@ -35,7 +35,7 @@ integral, center mass, average value
 
 open MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
 
-open Topology BigOperators Ennreal Convex
+open Topology BigOperators ENNReal Convex
 
 variable {α E F : Type _} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E]
   [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {μ : Measure α}
@@ -100,7 +100,7 @@ theorem average_eq' (f : α → E) : (⨍ x, f x ∂μ) = ∫ x, f x ∂(μ univ
 #align measure_theory.average_eq' MeasureTheory.average_eq'
 
 theorem average_eq (f : α → E) : (⨍ x, f x ∂μ) = (μ univ).toReal⁻¹ • ∫ x, f x ∂μ := by
-  rw [average_eq', integral_smul_measure, Ennreal.toReal_inv]
+  rw [average_eq', integral_smul_measure, ENNReal.toReal_inv]
 #align measure_theory.average_eq MeasureTheory.average_eq
 
 theorem average_eq_integral [IsProbabilityMeasure μ] (f : α → E) : (⨍ x, f x ∂μ) = ∫ x, f x ∂μ := by
@@ -114,7 +114,7 @@ theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
   cases' eq_or_ne μ 0 with hμ hμ
   · rw [hμ, integral_zero_measure, average_zero_measure, smul_zero]
   · rw [average_eq, smul_inv_smul₀]
-    refine' (Ennreal.toReal_pos _ <| measure_ne_top _ _).ne'
+    refine' (ENNReal.toReal_pos _ <| measure_ne_top _ _).ne'
     rwa [Ne.def, measure_univ_eq_zero]
 #align measure_theory.measure_smul_average MeasureTheory.measure_smul_average
 
@@ -141,7 +141,7 @@ theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeas
         ((ν univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂ν :=
   by
   simp only [div_eq_inv_mul, mul_smul, measure_smul_average, ← smul_add, ←
-    integral_add_measure hμ hν, ← Ennreal.toReal_add (measure_ne_top μ _) (measure_ne_top ν _)]
+    integral_add_measure hμ hν, ← ENNReal.toReal_add (measure_ne_top μ _) (measure_ne_top ν _)]
   rw [average_eq, measure.add_apply]
 #align measure_theory.average_add_measure MeasureTheory.average_add_measure
 
@@ -172,8 +172,8 @@ theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AeDisj
     (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
     (⨍ x in s ∪ t, f x ∂μ) ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in t, f x ∂μ) :=
   by
-  replace hs₀ : 0 < (μ s).toReal; exact Ennreal.toReal_pos hs₀ hsμ
-  replace ht₀ : 0 < (μ t).toReal; exact Ennreal.toReal_pos ht₀ htμ
+  replace hs₀ : 0 < (μ s).toReal; exact ENNReal.toReal_pos hs₀ hsμ
+  replace ht₀ : 0 < (μ t).toReal; exact ENNReal.toReal_pos ht₀ htμ
   refine'
     mem_open_segment_iff_div.mpr
       ⟨(μ s).toReal, (μ t).toReal, hs₀, ht₀, (average_union hd ht hsμ htμ hfs hft).symm⟩
@@ -190,11 +190,11 @@ theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AeDisjoint
     exact right_mem_segment _ _ _
   · refine'
       mem_segment_iff_div.mpr
-        ⟨(μ s).toReal, (μ t).toReal, Ennreal.toReal_nonneg, Ennreal.toReal_nonneg, _,
+        ⟨(μ s).toReal, (μ t).toReal, ENNReal.toReal_nonneg, ENNReal.toReal_nonneg, _,
           (average_union hd ht hsμ htμ hfs hft).symm⟩
     calc
-      0 < (μ s).toReal := Ennreal.toReal_pos hse hsμ
-      _ ≤ _ := le_add_of_nonneg_right Ennreal.toReal_nonneg
+      0 < (μ s).toReal := ENNReal.toReal_pos hse hsμ
+      _ ≤ _ := le_add_of_nonneg_right ENNReal.toReal_nonneg
       
 #align measure_theory.average_union_mem_segment MeasureTheory.average_union_mem_segment
 
@@ -209,16 +209,16 @@ theorem average_mem_openSegment_compl_self [IsFiniteMeasure μ] {f : α → E} {
 @[simp]
 theorem average_const [IsFiniteMeasure μ] [h : μ.ae.ne_bot] (c : E) : (⨍ x, c ∂μ) = c := by
   simp only [average_eq, integral_const, measure.restrict_apply, MeasurableSet.univ, one_smul,
-    univ_inter, smul_smul, ← Ennreal.toReal_inv, ← Ennreal.toReal_mul, Ennreal.inv_mul_cancel,
+    univ_inter, smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel,
     measure_ne_top μ univ, Ne.def, measure_univ_eq_zero, ae_ne_bot.1 h, not_false_iff,
-    Ennreal.one_toReal]
+    ENNReal.one_toReal]
 #align measure_theory.average_const MeasureTheory.average_const
 
 theorem set_average_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : E) :
     (⨍ x in s, c ∂μ) = c := by
   simp only [set_average_eq, integral_const, measure.restrict_apply, MeasurableSet.univ, univ_inter,
-    smul_smul, ← Ennreal.toReal_inv, ← Ennreal.toReal_mul, Ennreal.inv_mul_cancel hs₀ hs,
-    Ennreal.one_toReal, one_smul]
+    smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel hs₀ hs,
+    ENNReal.one_toReal, one_smul]
 #align measure_theory.set_average_const MeasureTheory.set_average_const
 
 end MeasureTheory

bump to nightly-2023-02-21-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

1107b979

Changes in mathlib4

mathlib3

mathlib4

chore: adapt to multiple goal linter 1 (#12338)

A PR accompanying #12339.

Zulip discussion

45f93f3f

Diff

@@ -695,7 +695,7 @@ theorem measure_setLaverage_le_pos (hμ : μ s ≠ 0) (hs : NullMeasurableSet s
   rintro x ⟨hfx, hx⟩
   dsimp at hfx
   rw [← toReal_laverage hg.aemeasurable, toReal_le_toReal (setLaverage_lt_top hint).ne hx] at hfx
-  exact hfx.trans (hgf _)
+  · exact hfx.trans (hgf _)
   · simp_rw [ae_iff, not_ne_iff]
     exact measure_eq_top_of_lintegral_ne_top hg.aemeasurable hint
 #align measure_theory.measure_set_laverage_le_pos MeasureTheory.measure_setLaverage_le_pos

chore: replace set_integral with setIntegral (#12215)

Done with a global search and replace, and then (to fix the #align lines), replace (#align \S*)setIntegral with $1set_integral.

f4b4298b

Diff

@@ -363,11 +363,11 @@ theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : ⨍ x, f x ∂μ = 
 #align measure_theory.average_congr MeasureTheory.average_congr
 
 theorem setAverage_congr (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ := by
-  simp only [setAverage_eq, set_integral_congr_set_ae h, measure_congr h]
+  simp only [setAverage_eq, setIntegral_congr_set_ae h, measure_congr h]
 #align measure_theory.set_average_congr MeasureTheory.setAverage_congr
 
 theorem setAverage_congr_fun (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
-    ⨍ x in s, f x ∂μ = ⨍ x in s, g x ∂μ := by simp only [average_eq, set_integral_congr_ae hs h]
+    ⨍ x in s, f x ∂μ = ⨍ x in s, g x ∂μ := by simp only [average_eq, setIntegral_congr_ae hs h]
 #align measure_theory.set_average_congr_fun MeasureTheory.setAverage_congr_fun
 
 theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E}
@@ -525,7 +525,7 @@ theorem measure_le_setAverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf
     exact AEStronglyMeasurable.nullMeasurableSet_le hf.1 aestronglyMeasurable_const
   haveI := Fact.mk hμ₁.lt_top
   refine' (integral_sub_average (μ.restrict s) f).not_gt _
-  refine' (set_integral_pos_iff_support_of_nonneg_ae _ _).2 _
+  refine' (setIntegral_pos_iff_support_of_nonneg_ae _ _).2 _
   · refine' eq_bot_mono (measure_mono fun x hx => _) H
     simp only [Pi.zero_apply, sub_nonneg, mem_compl_iff, mem_setOf_eq, not_le] at hx
     exact hx.le
@@ -863,7 +863,7 @@ theorem tendsto_integral_smul_of_tendsto_average_norm_sub
       intro x hx
       have : g i x = 0 := by rw [← Function.nmem_support]; exact fun h ↦ hx (hi h)
       simp [this]
-    rw [← set_integral_eq_integral_of_forall_compl_eq_zero this (μ := μ)]
+    rw [← setIntegral_eq_integral_of_forall_compl_eq_zero this (μ := μ)]
     refine' integral_mono_of_nonneg (eventually_of_forall (fun x ↦ by positivity)) _
       (eventually_of_forall (fun x ↦ _))
     · apply (Integrable.sub h''i _).norm.const_mul

chore: avoid Ne.def (adaptation for nightly-2024-03-27) (#11813)

08686b25

Diff

@@ -344,7 +344,7 @@ theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
   · rw [hμ, integral_zero_measure, average_zero_measure, smul_zero]
   · rw [average_eq, smul_inv_smul₀]
     refine' (ENNReal.toReal_pos _ <| measure_ne_top _ _).ne'
-    rwa [Ne.def, measure_univ_eq_zero]
+    rwa [Ne, measure_univ_eq_zero]
 #align measure_theory.measure_smul_average MeasureTheory.measure_smul_average
 
 theorem setAverage_eq (f : α → E) (s : Set α) :

chore: classify "simp can prove" porting notes (#11550)

Classifies by adding issue number #10618 to porting notes claiming "simp can prove it".

5befdcb9

Diff

@@ -446,7 +446,7 @@ theorem setAverage_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (
   have := NeZero.mk hs₀; have := Fact.mk hs.lt_top; average_const _ _
 #align measure_theory.set_average_const MeasureTheory.setAverage_const
 
--- Porting note: was `@[simp]` but `simp` can prove it
+-- Porting note (#10618): was `@[simp]` but `simp` can prove it
 theorem integral_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :
     ∫ _, ⨍ a, f a ∂μ ∂μ = ∫ x, f x ∂μ := by simp
 #align measure_theory.integral_average MeasureTheory.integral_average

style: homogenise porting notes (#11145)

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

converts "--porting note" into "-- Porting note";
converts "porting note" into "Porting note".

8be0b69c

Diff

@@ -242,7 +242,7 @@ theorem setLaverage_one (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) : ⨍⁻ _x in
   setLaverage_const hs₀ hs _
 #align measure_theory.set_laverage_one MeasureTheory.setLaverage_one
 
---porting note: Dropped `simp` because of `simp` seeing through `1 : α → ℝ≥0∞` and applying
+-- Porting note: Dropped `simp` because of `simp` seeing through `1 : α → ℝ≥0∞` and applying
 -- `lintegral_const`. This is suboptimal.
 theorem lintegral_laverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ≥0∞) :
     ∫⁻ _x, ⨍⁻ a, f a ∂μ ∂μ = ∫⁻ x, f x ∂μ := by
@@ -446,7 +446,7 @@ theorem setAverage_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (
   have := NeZero.mk hs₀; have := Fact.mk hs.lt_top; average_const _ _
 #align measure_theory.set_average_const MeasureTheory.setAverage_const
 
--- porting note: was `@[simp]` but `simp` can prove it
+-- Porting note: was `@[simp]` but `simp` can prove it
 theorem integral_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :
     ∫ _, ⨍ a, f a ∂μ ∂μ = ∫ x, f x ∂μ := by simp
 #align measure_theory.integral_average MeasureTheory.integral_average

chore: remove stream-of-consciousness uses of have, replace and suffices (#10640)

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

a13e8040

Diff

@@ -404,9 +404,9 @@ theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AEDisj
     (ht : NullMeasurableSet t μ) (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞)
     (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
     ⨍ x in s ∪ t, f x ∂μ ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in t, f x ∂μ) := by
-  replace hs₀ : 0 < (μ s).toReal; exact ENNReal.toReal_pos hs₀ hsμ
-  replace ht₀ : 0 < (μ t).toReal; exact ENNReal.toReal_pos ht₀ htμ
-  refine' mem_openSegment_iff_div.mpr
+  replace hs₀ : 0 < (μ s).toReal := ENNReal.toReal_pos hs₀ hsμ
+  replace ht₀ : 0 < (μ t).toReal := ENNReal.toReal_pos ht₀ htμ
+  exact mem_openSegment_iff_div.mpr
     ⟨(μ s).toReal, (μ t).toReal, hs₀, ht₀, (average_union hd ht hsμ htμ hfs hft).symm⟩
 #align measure_theory.average_union_mem_open_segment MeasureTheory.average_union_mem_openSegment
 
@@ -520,8 +520,8 @@ measure. -/
 theorem measure_le_setAverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
     0 < μ ({x ∈ s | f x ≤ ⨍ a in s, f a ∂μ}) := by
   refine' pos_iff_ne_zero.2 fun H => _
-  replace H : (μ.restrict s) {x | f x ≤ ⨍ a in s, f a ∂μ} = 0
-  · rwa [restrict_apply₀, inter_comm]
+  replace H : (μ.restrict s) {x | f x ≤ ⨍ a in s, f a ∂μ} = 0 := by
+    rwa [restrict_apply₀, inter_comm]
     exact AEStronglyMeasurable.nullMeasurableSet_le hf.1 aestronglyMeasurable_const
   haveI := Fact.mk hμ₁.lt_top
   refine' (integral_sub_average (μ.restrict s) f).not_gt _

doc: @[inherit_doc] on notations (#9942)

Make all the notations that unambiguously should inherit the docstring of their definition actually inherit it.

Also write a few docstrings by hand. I only wrote the ones I was competent to write and which I was sure of. Some docstrings come from mathlib3 as they were lost during the early port.

This PR is only intended as a first pass There are many more docstrings to add.

08f7e99d

Diff

@@ -62,18 +62,45 @@ namespace MeasureTheory
 section ENNReal
 variable (μ) {f g : α → ℝ≥0∞}
 
-/-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. a measure `μ`, notation: `⨍⁻ x, f x ∂μ`.
-It is defined as `μ univ⁻¹ * ∫⁻ x, f x ∂μ`, so it is equal to zero if `μ` is an infinite measure. If
+/-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. a measure `μ`, denoted `⨍⁻ x, f x ∂μ`.
+
+It is equal to `(μ univ)⁻¹ * ∫⁻ x, f x ∂μ`, so it takes value zero if `μ` is an infinite measure. If
 `μ` is a probability measure, then the average of any function is equal to its integral.
 
-For the average on a set, use `⨍⁻ x in s, f x ∂μ` (defined as `⨍⁻ x, f x ∂(μ.restrict s)`). For
+For the average on a set, use `⨍⁻ x in s, f x ∂μ`, defined as `⨍⁻ x, f x ∂(μ.restrict s)`. For the
 average w.r.t. the volume, one can omit `∂volume`. -/
 noncomputable def laverage (f : α → ℝ≥0∞) := ∫⁻ x, f x ∂(μ univ)⁻¹ • μ
 #align measure_theory.laverage MeasureTheory.laverage
 
-notation3"⨍⁻ "(...)", "r:60:(scoped f => f)" ∂"μ:70 => laverage μ r
-notation3"⨍⁻ "(...)", "r:60:(scoped f => laverage volume f) => r
-notation3"⨍⁻ "(...)" in "s", "r:60:(scoped f => f)" ∂"μ:70 => laverage (Measure.restrict μ s) r
+/-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. a measure `μ`.
+
+It is equal to `(μ univ)⁻¹ * ∫⁻ x, f x ∂μ`, so it takes value zero if `μ` is an infinite measure. If
+`μ` is a probability measure, then the average of any function is equal to its integral.
+
+For the average on a set, use `⨍⁻ x in s, f x ∂μ`, defined as `⨍⁻ x, f x ∂(μ.restrict s)`. For the
+average w.r.t. the volume, one can omit `∂volume`. -/
+notation3 "⨍⁻ "(...)", "r:60:(scoped f => f)" ∂"μ:70 => laverage μ r
+
+/-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. to the standard measure.
+
+It is equal to `(volume univ)⁻¹ * ∫⁻ x, f x`, so it takes value zero if the space has infinite
+measure. In a probability space, the average of any function is equal to its integral.
+
+For the average on a set, use `⨍⁻ x in s, f x`, defined as `⨍⁻ x, f x ∂(volume.restrict s)`. -/
+notation3 "⨍⁻ "(...)", "r:60:(scoped f => laverage volume f) => r
+
+/-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. a measure `μ` on a set `s`.
+
+It is equal to `(μ s)⁻¹ * ∫⁻ x, f x ∂μ`, so it takes value zero if `s` has infinite measure. If `s`
+has measure `1`, then the average of any function is equal to its integral.
+
+For the average w.r.t. the volume, one can omit `∂volume`. -/
+notation3 "⨍⁻ "(...)" in "s", "r:60:(scoped f => f)" ∂"μ:70 => laverage (Measure.restrict μ s) r
+
+/-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. to the standard measure on a set `s`.
+
+It is equal to `(volume s)⁻¹ * ∫⁻ x, f x`, so it takes value zero if `s` has infinite measure. If
+`s` has measure `1`, then the average of any function is equal to its integral. -/
 notation3 (prettyPrint := false)
   "⨍⁻ "(...)" in "s", "r:60:(scoped f => laverage Measure.restrict volume s f) => r
 
@@ -237,27 +264,51 @@ section NormedAddCommGroup
 variable (μ)
 variable {f g : α → E}
 
-/-- Average value of a function `f` w.r.t. a measure `μ`, notation: `⨍ x, f x ∂μ`. It is defined as
-`(μ univ).toReal⁻¹ • ∫ x, f x ∂μ`, so it is equal to zero if `f` is not integrable or if `μ` is an
-infinite measure. If `μ` is a probability measure, then the average of any function is equal to its
-integral.
+/-- Average value of a function `f` w.r.t. a measure `μ`, denoted `⨍ x, f x ∂μ`.
+
+It is equal to `(μ univ).toReal⁻¹ • ∫ x, f x ∂μ`, so it takes value zero if `f` is not integrable or
+if `μ` is an infinite measure. If `μ` is a probability measure, then the average of any function is
+equal to its integral.
 
-For the average on a set, use `⨍ x in s, f x ∂μ` (defined as `⨍ x, f x ∂(μ.restrict s)`). For
+For the average on a set, use `⨍ x in s, f x ∂μ`, defined as `⨍ x, f x ∂(μ.restrict s)`. For the
 average w.r.t. the volume, one can omit `∂volume`. -/
 noncomputable def average (f : α → E) :=
   ∫ x, f x ∂(μ univ)⁻¹ • μ
 #align measure_theory.average MeasureTheory.average
 
-@[inherit_doc average]
+/-- Average value of a function `f` w.r.t. a measure `μ`.
+
+It is equal to `(μ univ).toReal⁻¹ • ∫ x, f x ∂μ`, so it takes value zero if `f` is not integrable or
+if `μ` is an infinite measure. If `μ` is a probability measure, then the average of any function is
+equal to its integral.
+
+For the average on a set, use `⨍ x in s, f x ∂μ`, defined as `⨍ x, f x ∂(μ.restrict s)`. For the
+average w.r.t. the volume, one can omit `∂volume`. -/
 notation3 "⨍ "(...)", "r:60:(scoped f => f)" ∂"μ:70 => average μ r
 
-@[inherit_doc average]
+/-- Average value of a function `f` w.r.t. to the standard measure.
+
+It is equal to `(volume univ).toReal⁻¹ * ∫ x, f x`, so it takes value zero if `f` is not integrable
+or if the space has infinite measure. In a probability space, the average of any function is equal
+to its integral.
+
+For the average on a set, use `⨍ x in s, f x`, defined as `⨍ x, f x ∂(volume.restrict s)`. -/
 notation3 "⨍ "(...)", "r:60:(scoped f => average volume f) => r
 
-@[inherit_doc average]
+/-- Average value of a function `f` w.r.t. a measure `μ` on a set `s`.
+
+It is equal to `(μ s).toReal⁻¹ * ∫ x, f x ∂μ`, so it takes value zero if `f` is not integrable on
+`s` or if `s` has infinite measure. If `s` has measure `1`, then the average of any function is
+equal to its integral.
+
+For the average w.r.t. the volume, one can omit `∂volume`. -/
 notation3 "⨍ "(...)" in "s", "r:60:(scoped f => f)" ∂"μ:70 => average (Measure.restrict μ s) r
 
-@[inherit_doc average]
+/-- Average value of a function `f` w.r.t. to the standard measure on a set `s`.
+
+It is equal to `(volume s).toReal⁻¹ * ∫ x, f x`, so it takes value zero `f` is not integrable on `s`
+or if `s` has infinite measure. If `s` has measure `1`, then the average of any function is equal to
+its integral. -/
 notation3 "⨍ "(...)" in "s", "r:60:(scoped f => average (Measure.restrict volume s) f) => r
 
 @[simp]

chore: remove uses of cases' (#9171)

I literally went through and regex'd some uses of cases', replacing them with rcases; this is meant to be a low effort PR as I hope that tools can do this in the future.

rcases is an easier replacement than cases, though with better tools we could in future do a second pass converting simple rcases added here (and existing ones) to cases.

3fca282c

Diff

@@ -99,7 +99,7 @@ theorem laverage_eq_lintegral [IsProbabilityMeasure μ] (f : α → ℝ≥0∞)
 @[simp]
 theorem measure_mul_laverage [IsFiniteMeasure μ] (f : α → ℝ≥0∞) :
     μ univ * ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ := by
-  cases' eq_or_ne μ 0 with hμ hμ
+  rcases eq_or_ne μ 0 with hμ | hμ
   · rw [hμ, lintegral_zero_measure, laverage_zero_measure, mul_zero]
   · rw [laverage_eq, ENNReal.mul_div_cancel' (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)]
 #align measure_theory.measure_mul_laverage MeasureTheory.measure_mul_laverage
@@ -289,7 +289,7 @@ theorem average_eq_integral [IsProbabilityMeasure μ] (f : α → E) : ⨍ x, f
 @[simp]
 theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
     (μ univ).toReal • ⨍ x, f x ∂μ = ∫ x, f x ∂μ := by
-  cases' eq_or_ne μ 0 with hμ hμ
+  rcases eq_or_ne μ 0 with hμ | hμ
   · rw [hμ, integral_zero_measure, average_zero_measure, smul_zero]
   · rw [average_eq, smul_inv_smul₀]
     refine' (ENNReal.toReal_pos _ <| measure_ne_top _ _).ne'

chore: space after ← (#8178)

Co-authored-by: Moritz Firsching <firsching@google.com>

5fb9beab

Diff

@@ -149,14 +149,14 @@ theorem laverage_add_measure :
   · rw [not_isFiniteMeasure_iff] at hν
     simp [laverage_eq, hν]
   haveI := hμ; haveI := hν
-  simp only [←ENNReal.mul_div_right_comm, measure_mul_laverage, ←ENNReal.add_div,
-    ←lintegral_add_measure, ←Measure.add_apply, ←laverage_eq]
+  simp only [← ENNReal.mul_div_right_comm, measure_mul_laverage, ← ENNReal.add_div,
+    ← lintegral_add_measure, ← Measure.add_apply, ← laverage_eq]
 #align measure_theory.laverage_add_measure MeasureTheory.laverage_add_measure
 
 theorem measure_mul_setLaverage (f : α → ℝ≥0∞) (h : μ s ≠ ∞) :
     μ s * ⨍⁻ x in s, f x ∂μ = ∫⁻ x in s, f x ∂μ := by
   have := Fact.mk h.lt_top
-  rw [←measure_mul_laverage, restrict_apply_univ]
+  rw [← measure_mul_laverage, restrict_apply_univ]
 #align measure_theory.measure_mul_set_laverage MeasureTheory.measure_mul_setLaverage
 
 theorem laverage_union (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ) :
@@ -171,7 +171,7 @@ theorem laverage_union_mem_openSegment (hd : AEDisjoint μ s t) (ht : NullMeasur
   refine'
     ⟨μ s / (μ s + μ t), μ t / (μ s + μ t), ENNReal.div_pos hs₀ <| add_ne_top.2 ⟨hsμ, htμ⟩,
       ENNReal.div_pos ht₀ <| add_ne_top.2 ⟨hsμ, htμ⟩, _, (laverage_union hd ht).symm⟩
-  rw [←ENNReal.add_div,
+  rw [← ENNReal.add_div,
     ENNReal.div_self (add_eq_zero.not.2 fun h => hs₀ h.1) (add_ne_top.2 ⟨hsμ, htμ⟩)]
 #align measure_theory.laverage_union_mem_open_segment MeasureTheory.laverage_union_mem_openSegment
 
@@ -179,12 +179,12 @@ theorem laverage_union_mem_segment (hd : AEDisjoint μ s t) (ht : NullMeasurable
     (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) :
     ⨍⁻ x in s ∪ t, f x ∂μ ∈ [⨍⁻ x in s, f x ∂μ -[ℝ≥0∞] ⨍⁻ x in t, f x ∂μ] := by
   by_cases hs₀ : μ s = 0
-  · rw [←ae_eq_empty] at hs₀
+  · rw [← ae_eq_empty] at hs₀
     rw [restrict_congr_set (hs₀.union EventuallyEq.rfl), empty_union]
     exact right_mem_segment _ _ _
   · refine'
       ⟨μ s / (μ s + μ t), μ t / (μ s + μ t), zero_le _, zero_le _, _, (laverage_union hd ht).symm⟩
-    rw [←ENNReal.add_div,
+    rw [← ENNReal.add_div,
       ENNReal.div_self (add_eq_zero.not.2 fun h => hs₀ h.1) (add_ne_top.2 ⟨hsμ, htμ⟩)]
 #align measure_theory.laverage_union_mem_segment MeasureTheory.laverage_union_mem_segment
 
@@ -324,8 +324,8 @@ theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeas
     ⨍ x, f x ∂(μ + ν) =
       ((μ univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂μ +
         ((ν univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂ν := by
-  simp only [div_eq_inv_mul, mul_smul, measure_smul_average, ←smul_add,
-    ←integral_add_measure hμ hν, ←ENNReal.toReal_add (measure_ne_top μ _) (measure_ne_top ν _)]
+  simp only [div_eq_inv_mul, mul_smul, measure_smul_average, ← smul_add,
+    ← integral_add_measure hμ hν, ← ENNReal.toReal_add (measure_ne_top μ _) (measure_ne_top ν _)]
   rw [average_eq, Measure.add_apply]
 #align measure_theory.average_add_measure MeasureTheory.average_add_measure
 
@@ -337,7 +337,7 @@ theorem average_pair {f : α → E} {g : α → F} (hfi : Integrable f μ) (hgi
 theorem measure_smul_setAverage (f : α → E) {s : Set α} (h : μ s ≠ ∞) :
     (μ s).toReal • ⨍ x in s, f x ∂μ = ∫ x in s, f x ∂μ := by
   haveI := Fact.mk h.lt_top
-  rw [←measure_smul_average, restrict_apply_univ]
+  rw [← measure_smul_average, restrict_apply_univ]
 #align measure_theory.measure_smul_set_average MeasureTheory.measure_smul_setAverage
 
 theorem average_union {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
@@ -364,7 +364,7 @@ theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint
     (hft : IntegrableOn f t μ) :
     ⨍ x in s ∪ t, f x ∂μ ∈ [⨍ x in s, f x ∂μ -[ℝ] ⨍ x in t, f x ∂μ] := by
   by_cases hse : μ s = 0
-  · rw [←ae_eq_empty] at hse
+  · rw [← ae_eq_empty] at hse
     rw [restrict_congr_set (hse.union EventuallyEq.rfl), empty_union]
     exact right_mem_segment _ _ _
   · refine'
@@ -613,15 +613,15 @@ theorem measure_le_setLaverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞)
   obtain h | h := eq_or_ne (∫⁻ a in s, f a ∂μ) ∞
   · simpa [mul_top, hμ₁, laverage, h, top_div_of_ne_top hμ₁, pos_iff_ne_zero] using hμ
   have := measure_le_setAverage_pos hμ hμ₁ (integrable_toReal_of_lintegral_ne_top hf h)
-  rw [←setOf_inter_eq_sep, ←Measure.restrict_apply₀
+  rw [← setOf_inter_eq_sep, ← Measure.restrict_apply₀
     (hf.aestronglyMeasurable.nullMeasurableSet_le aestronglyMeasurable_const)]
-  rw [←setOf_inter_eq_sep, ←Measure.restrict_apply₀
+  rw [← setOf_inter_eq_sep, ← Measure.restrict_apply₀
     (hf.ennreal_toReal.aestronglyMeasurable.nullMeasurableSet_le aestronglyMeasurable_const),
-    ←measure_diff_null (measure_eq_top_of_lintegral_ne_top hf h)] at this
+    ← measure_diff_null (measure_eq_top_of_lintegral_ne_top hf h)] at this
   refine' this.trans_le (measure_mono _)
   rintro x ⟨hfx, hx⟩
   dsimp at hfx
-  rwa [←toReal_laverage hf, toReal_le_toReal hx (setLaverage_lt_top h).ne] at hfx
+  rwa [← toReal_laverage hf, toReal_le_toReal hx (setLaverage_lt_top h).ne] at hfx
   · simp_rw [ae_iff, not_ne_iff]
     exact measure_eq_top_of_lintegral_ne_top hf h
 #align measure_theory.measure_le_set_laverage_pos MeasureTheory.measure_le_setLaverage_pos
@@ -637,13 +637,13 @@ theorem measure_setLaverage_le_pos (hμ : μ s ≠ 0) (hs : NullMeasurableSet s
   rw [hfg] at hint
   have :=
     measure_setAverage_le_pos hμ hμ₁ (integrable_toReal_of_lintegral_ne_top hg.aemeasurable hint)
-  simp_rw [←setOf_inter_eq_sep, ←Measure.restrict_apply₀' hs, hfg']
-  rw [←setOf_inter_eq_sep, ←Measure.restrict_apply₀' hs, ←
+  simp_rw [← setOf_inter_eq_sep, ← Measure.restrict_apply₀' hs, hfg']
+  rw [← setOf_inter_eq_sep, ← Measure.restrict_apply₀' hs, ←
     measure_diff_null (measure_eq_top_of_lintegral_ne_top hg.aemeasurable hint)] at this
   refine' this.trans_le (measure_mono _)
   rintro x ⟨hfx, hx⟩
   dsimp at hfx
-  rw [←toReal_laverage hg.aemeasurable, toReal_le_toReal (setLaverage_lt_top hint).ne hx] at hfx
+  rw [← toReal_laverage hg.aemeasurable, toReal_le_toReal (setLaverage_lt_top hint).ne hx] at hfx
   exact hfx.trans (hgf _)
   · simp_rw [ae_iff, not_ne_iff]
     exact measure_eq_top_of_lintegral_ne_top hg.aemeasurable hint
@@ -682,7 +682,7 @@ avoiding a null set. -/
 theorem exists_not_mem_null_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a : α, f a ∂μ ≠ ∞) (hN : μ N = 0) :
     ∃ x, x ∉ N ∧ ⨍⁻ a, f a ∂μ ≤ f x := by
   have := measure_laverage_le_pos hμ hint
-  rw [←measure_diff_null hN] at this
+  rw [← measure_diff_null hN] at this
   obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
   exact ⟨x, hxN, hx⟩
 #align measure_theory.exists_not_mem_null_laverage_le MeasureTheory.exists_not_mem_null_laverage_le
@@ -709,7 +709,7 @@ avoiding a null set. -/
 theorem exists_not_mem_null_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) (hN : μ N = 0) :
     ∃ x, x ∉ N ∧ f x ≤ ⨍⁻ a, f a ∂μ := by
   have := measure_le_laverage_pos hμ hf
-  rw [←measure_diff_null hN] at this
+  rw [← measure_diff_null hN] at this
   obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
   exact ⟨x, hxN, hx⟩
 #align measure_theory.exists_not_mem_null_le_laverage MeasureTheory.exists_not_mem_null_le_laverage

chore: tidy various files (#7017)

a8cb5022

Diff

@@ -603,10 +603,10 @@ theorem exists_not_mem_null_integral_le (hf : Integrable f μ) (hN : μ N = 0) :
 end ProbabilityMeasure
 end FirstMomentReal
 
-section FirstMomentEnnreal
+section FirstMomentENNReal
 variable {N : Set α} {f : α → ℝ≥0∞}
 
-/-- **First moment method**. A measurable function is smaller than its mean on a set of ositive
+/-- **First moment method**. A measurable function is smaller than its mean on a set of positive
 measure. -/
 theorem measure_le_setLaverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞)
     (hf : AEMeasurable f (μ.restrict s)) : 0 < μ {x ∈ s | f x ≤ ⨍⁻ a in s, f a ∂μ} := by
@@ -626,7 +626,7 @@ theorem measure_le_setLaverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞)
     exact measure_eq_top_of_lintegral_ne_top hf h
 #align measure_theory.measure_le_set_laverage_pos MeasureTheory.measure_le_setLaverage_pos
 
-/-- **First moment method**. A measurable function is greater than its mean on a set of ositive
+/-- **First moment method**. A measurable function is greater than its mean on a set of positive
 measure. -/
 theorem measure_setLaverage_le_pos (hμ : μ s ≠ 0) (hs : NullMeasurableSet s μ)
     (hint : ∫⁻ a in s, f a ∂μ ≠ ∞) : 0 < μ {x ∈ s | ⨍⁻ a in s, f a ∂μ ≤ f x} := by
@@ -649,21 +649,21 @@ theorem measure_setLaverage_le_pos (hμ : μ s ≠ 0) (hs : NullMeasurableSet s
     exact measure_eq_top_of_lintegral_ne_top hg.aemeasurable hint
 #align measure_theory.measure_set_laverage_le_pos MeasureTheory.measure_setLaverage_le_pos
 
-/-- **First moment method**. The minimum of a measurable function is smaller than its ean. -/
+/-- **First moment method**. The minimum of a measurable function is smaller than its mean. -/
 theorem exists_le_setLaverage (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : AEMeasurable f (μ.restrict s)) :
     ∃ x ∈ s, f x ≤ ⨍⁻ a in s, f a ∂μ :=
   let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_le_setLaverage_pos hμ hμ₁ hf).ne'
   ⟨x, hx, h⟩
 #align measure_theory.exists_le_set_laverage MeasureTheory.exists_le_setLaverage
 
-/-- **First moment method**. The maximum of a measurable function is greater than its ean. -/
+/-- **First moment method**. The maximum of a measurable function is greater than its mean. -/
 theorem exists_setLaverage_le (hμ : μ s ≠ 0) (hs : NullMeasurableSet s μ)
     (hint : ∫⁻ a in s, f a ∂μ ≠ ∞) : ∃ x ∈ s, ⨍⁻ a in s, f a ∂μ ≤ f x :=
   let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_setLaverage_le_pos hμ hs hint).ne'
   ⟨x, hx, h⟩
 #align measure_theory.exists_set_laverage_le MeasureTheory.exists_setLaverage_le
 
-/-- **First moment method**. A measurable function is greater than its mean on a set of ositive
+/-- **First moment method**. A measurable function is greater than its mean on a set of positive
 measure. -/
 theorem measure_laverage_le_pos (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞) :
     0 < μ {x | ⨍⁻ a, f a ∂μ ≤ f x} := by
@@ -671,13 +671,13 @@ theorem measure_laverage_le_pos (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠
     @measure_setLaverage_le_pos _ _ _ _ f (measure_univ_ne_zero.2 hμ) nullMeasurableSet_univ
 #align measure_theory.measure_laverage_le_pos MeasureTheory.measure_laverage_le_pos
 
-/-- **First moment method**. The maximum of a measurable function is greater than its ean. -/
+/-- **First moment method**. The maximum of a measurable function is greater than its mean. -/
 theorem exists_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞) : ∃ x, ⨍⁻ a, f a ∂μ ≤ f x :=
   let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_laverage_le_pos hμ hint).ne'
   ⟨x, hx⟩
 #align measure_theory.exists_laverage_le MeasureTheory.exists_laverage_le
 
-/-- **First moment method**. The maximum of a measurable function is greater than its ean, while
+/-- **First moment method**. The maximum of a measurable function is greater than its mean, while
 avoiding a null set. -/
 theorem exists_not_mem_null_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a : α, f a ∂μ ≠ ∞) (hN : μ N = 0) :
     ∃ x, x ∉ N ∧ ⨍⁻ a, f a ∂μ ≤ f x := by
@@ -690,7 +690,7 @@ theorem exists_not_mem_null_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a : α,
 section FiniteMeasure
 variable [IsFiniteMeasure μ]
 
-/-- **First moment method**. A measurable function is smaller than its mean on a set of ositive
+/-- **First moment method**. A measurable function is smaller than its mean on a set of positive
 measure. -/
 theorem measure_le_laverage_pos (hμ : μ ≠ 0) (hf : AEMeasurable f μ) :
     0 < μ {x | f x ≤ ⨍⁻ a, f a ∂μ} := by
@@ -698,13 +698,13 @@ theorem measure_le_laverage_pos (hμ : μ ≠ 0) (hf : AEMeasurable f μ) :
     measure_le_setLaverage_pos (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _) hf.restrict
 #align measure_theory.measure_le_laverage_pos MeasureTheory.measure_le_laverage_pos
 
-/-- **First moment method**. The minimum of a measurable function is smaller than its ean. -/
+/-- **First moment method**. The minimum of a measurable function is smaller than its mean. -/
 theorem exists_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) : ∃ x, f x ≤ ⨍⁻ a, f a ∂μ :=
   let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_le_laverage_pos hμ hf).ne'
   ⟨x, hx⟩
 #align measure_theory.exists_le_laverage MeasureTheory.exists_le_laverage
 
-/-- **First moment method**. The minimum of a measurable function is smaller than its ean, while
+/-- **First moment method**. The minimum of a measurable function is smaller than its mean, while
 avoiding a null set. -/
 theorem exists_not_mem_null_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) (hN : μ N = 0) :
     ∃ x, x ∉ N ∧ f x ≤ ⨍⁻ a, f a ∂μ := by
@@ -734,18 +734,18 @@ theorem measure_lintegral_le_pos (hint : ∫⁻ a, f a ∂μ ≠ ∞) : 0 < μ {
     measure_laverage_le_pos (IsProbabilityMeasure.ne_zero μ) hint
 #align measure_theory.measure_lintegral_le_pos MeasureTheory.measure_lintegral_le_pos
 
-/-- **First moment method**. The minimum of a measurable function is smaller than its ntegral. -/
+/-- **First moment method**. The minimum of a measurable function is smaller than its integral. -/
 theorem exists_le_lintegral (hf : AEMeasurable f μ) : ∃ x, f x ≤ ∫⁻ a, f a ∂μ := by
   simpa only [laverage_eq_lintegral] using exists_le_laverage (IsProbabilityMeasure.ne_zero μ) hf
 #align measure_theory.exists_le_lintegral MeasureTheory.exists_le_lintegral
 
-/-- **First moment method**. The maximum of a measurable function is greater than its ntegral. -/
+/-- **First moment method**. The maximum of a measurable function is greater than its integral. -/
 theorem exists_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) : ∃ x, ∫⁻ a, f a ∂μ ≤ f x := by
   simpa only [laverage_eq_lintegral] using
     exists_laverage_le (IsProbabilityMeasure.ne_zero μ) hint
 #align measure_theory.exists_lintegral_le MeasureTheory.exists_lintegral_le
 
-/-- **First moment method**. The minimum of a measurable function is smaller than its ntegral,
+/-- **First moment method**. The minimum of a measurable function is smaller than its integral,
 while avoiding a null set. -/
 theorem exists_not_mem_null_le_lintegral (hf : AEMeasurable f μ) (hN : μ N = 0) :
     ∃ x, x ∉ N ∧ f x ≤ ∫⁻ a, f a ∂μ := by
@@ -753,7 +753,7 @@ theorem exists_not_mem_null_le_lintegral (hf : AEMeasurable f μ) (hN : μ N = 0
     exists_not_mem_null_le_laverage (IsProbabilityMeasure.ne_zero μ) hf hN
 #align measure_theory.exists_not_mem_null_le_lintegral MeasureTheory.exists_not_mem_null_le_lintegral
 
-/-- **First moment method**. The maximum of a measurable function is greater than its ntegral,
+/-- **First moment method**. The maximum of a measurable function is greater than its integral,
 while avoiding a null set. -/
 theorem exists_not_mem_null_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) (hN : μ N = 0) :
     ∃ x, x ∉ N ∧ ∫⁻ a, f a ∂μ ≤ f x := by
@@ -762,7 +762,7 @@ theorem exists_not_mem_null_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) (h
 #align measure_theory.exists_not_mem_null_lintegral_le MeasureTheory.exists_not_mem_null_lintegral_le
 
 end ProbabilityMeasure
-end FirstMomentEnnreal
+end FirstMomentENNReal
 
 /-- If the average of a function `f` along a sequence of sets `aₙ` converges to `c` (more precisely,
 we require that `⨍ y in a i, ‖f y - c‖ ∂μ` tends to `0`), then the integral of `gₙ • f` also tends

chore: drop MulZeroClass. in mul_zero/zero_mul (#6682)

Search&replace MulZeroClass.mul_zero -> mul_zero, MulZeroClass.zero_mul -> zero_mul.

These were introduced by Mathport, as the full name of mul_zero is actually MulZeroClass.mul_zero (it's exported with the short name).

277dea95

Diff

@@ -100,7 +100,7 @@ theorem laverage_eq_lintegral [IsProbabilityMeasure μ] (f : α → ℝ≥0∞)
 theorem measure_mul_laverage [IsFiniteMeasure μ] (f : α → ℝ≥0∞) :
     μ univ * ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ := by
   cases' eq_or_ne μ 0 with hμ hμ
-  · rw [hμ, lintegral_zero_measure, laverage_zero_measure, MulZeroClass.mul_zero]
+  · rw [hμ, lintegral_zero_measure, laverage_zero_measure, mul_zero]
   · rw [laverage_eq, ENNReal.mul_div_cancel' (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)]
 #align measure_theory.measure_mul_laverage MeasureTheory.measure_mul_laverage

chore: remove unused simps (#6632)

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

916c75c1

Diff

@@ -782,7 +782,6 @@ theorem tendsto_integral_smul_of_tendsto_average_norm_sub
     simp only [integral_undef hi, lt_self_iff_false, not_false_eq_true]
   have I : ∀ᶠ i in l, ∫ y, g i y • (f y - c) ∂μ + (∫ y, g i y ∂μ) • c = ∫ y, g i y • f y ∂μ := by
     filter_upwards [f_int, g_int, g_supp, g_bound] with i hif hig hisupp hibound
-    dsimp
     rw [← integral_smul_const, ← integral_add]
     · simp only [smul_sub, sub_add_cancel]
     · simp_rw [smul_sub]

chore: banish Type _ and Sort _ (#6499)

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

32de3f67

Diff

@@ -44,7 +44,7 @@ open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSp
 
 open scoped Topology BigOperators ENNReal Convex
 
-variable {α E F : Type _} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E]
+variable {α E F : Type*} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E]
   [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {μ ν : Measure α}
   {s t : Set α}
 
@@ -769,7 +769,7 @@ we require that `⨍ y in a i, ‖f y - c‖ ∂μ` tends to `0`), then the inte
 to `c` if `gₙ` is supported in `aₙ`, has integral converging to one and supremum at most `K / μ aₙ`.
 -/
 theorem tendsto_integral_smul_of_tendsto_average_norm_sub
-    {ι : Type _} {a : ι → Set α} {l : Filter ι} {f : α → E} {c : E} {g : ι → α → ℝ} (K : ℝ)
+    {ι : Type*} {a : ι → Set α} {l : Filter ι} {f : α → E} {c : E} {g : ι → α → ℝ} (K : ℝ)
     (hf : Tendsto (fun i ↦ ⨍ y in a i, ‖f y - c‖ ∂μ) l (𝓝 0))
     (f_int : ∀ᶠ i in l, IntegrableOn f (a i) μ)
     (hg : Tendsto (fun i ↦ ∫ y, g i y ∂μ) l (𝓝 1))

feat: The convolution of a locally integrable function f with a sequence of bump functions converges ae to f (#6102)

33baf7d8

Diff

@@ -763,4 +763,65 @@ theorem exists_not_mem_null_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) (h
 
 end ProbabilityMeasure
 end FirstMomentEnnreal
+
+/-- If the average of a function `f` along a sequence of sets `aₙ` converges to `c` (more precisely,
+we require that `⨍ y in a i, ‖f y - c‖ ∂μ` tends to `0`), then the integral of `gₙ • f` also tends
+to `c` if `gₙ` is supported in `aₙ`, has integral converging to one and supremum at most `K / μ aₙ`.
+-/
+theorem tendsto_integral_smul_of_tendsto_average_norm_sub
+    {ι : Type _} {a : ι → Set α} {l : Filter ι} {f : α → E} {c : E} {g : ι → α → ℝ} (K : ℝ)
+    (hf : Tendsto (fun i ↦ ⨍ y in a i, ‖f y - c‖ ∂μ) l (𝓝 0))
+    (f_int : ∀ᶠ i in l, IntegrableOn f (a i) μ)
+    (hg : Tendsto (fun i ↦ ∫ y, g i y ∂μ) l (𝓝 1))
+    (g_supp : ∀ᶠ i in l, Function.support (g i) ⊆ a i)
+    (g_bound : ∀ᶠ i in l, ∀ x, |g i x| ≤ K / (μ (a i)).toReal) :
+    Tendsto (fun i ↦ ∫ y, g i y • f y ∂μ) l (𝓝 c) := by
+  have g_int : ∀ᶠ i in l, Integrable (g i) μ := by
+    filter_upwards [(tendsto_order.1 hg).1 _ zero_lt_one] with i hi
+    contrapose hi
+    simp only [integral_undef hi, lt_self_iff_false, not_false_eq_true]
+  have I : ∀ᶠ i in l, ∫ y, g i y • (f y - c) ∂μ + (∫ y, g i y ∂μ) • c = ∫ y, g i y • f y ∂μ := by
+    filter_upwards [f_int, g_int, g_supp, g_bound] with i hif hig hisupp hibound
+    dsimp
+    rw [← integral_smul_const, ← integral_add]
+    · simp only [smul_sub, sub_add_cancel]
+    · simp_rw [smul_sub]
+      apply Integrable.sub _ (hig.smul_const _)
+      have A : Function.support (fun y ↦ g i y • f y) ⊆ a i := by
+        apply Subset.trans _ hisupp
+        exact Function.support_smul_subset_left _ _
+      rw [← integrableOn_iff_integrable_of_support_subset A]
+      apply Integrable.smul_of_top_right hif
+      exact memℒp_top_of_bound hig.aestronglyMeasurable.restrict
+        (K / (μ (a i)).toReal) (eventually_of_forall hibound)
+    · exact hig.smul_const _
+  have L0 : Tendsto (fun i ↦ ∫ y, g i y • (f y - c) ∂μ) l (𝓝 0) := by
+    have := hf.const_mul K
+    simp only [mul_zero] at this
+    refine' squeeze_zero_norm' _ this
+    filter_upwards [g_supp, g_bound, f_int, (tendsto_order.1 hg).1 _ zero_lt_one]
+      with i hi h'i h''i hi_int
+    have mu_ai : μ (a i) < ∞ := by
+      rw [lt_top_iff_ne_top]
+      intro h
+      simp only [h, ENNReal.top_toReal, _root_.div_zero, abs_nonpos_iff] at h'i
+      have : ∫ (y : α), g i y ∂μ = ∫ (y : α), 0 ∂μ := by congr; ext y; exact h'i y
+      simp [this] at hi_int
+    apply (norm_integral_le_integral_norm _).trans
+    simp_rw [average_eq, smul_eq_mul, ← integral_mul_left, norm_smul, ← mul_assoc, ← div_eq_mul_inv]
+    have : ∀ x, x ∉ a i → ‖g i x‖ * ‖(f x - c)‖ = 0 := by
+      intro x hx
+      have : g i x = 0 := by rw [← Function.nmem_support]; exact fun h ↦ hx (hi h)
+      simp [this]
+    rw [← set_integral_eq_integral_of_forall_compl_eq_zero this (μ := μ)]
+    refine' integral_mono_of_nonneg (eventually_of_forall (fun x ↦ by positivity)) _
+      (eventually_of_forall (fun x ↦ _))
+    · apply (Integrable.sub h''i _).norm.const_mul
+      change IntegrableOn (fun _ ↦ c) (a i) μ
+      simp [integrableOn_const, mu_ai]
+    · dsimp; gcongr; simpa using h'i x
+  have := L0.add (hg.smul_const c)
+  simp only [one_smul, zero_add] at this
+  exact Tendsto.congr' I this
+
 end MeasureTheory

refactor: use NeZero for measures (#6048)

Assume NeZero μ instead of μ.ae.NeBot everywhere, and sometimes instead of μ ≠ 0.

API changes

Convex.average_mem, Convex.set_average_mem, ConvexOn.average_mem_epigraph, ConcaveOn.average_mem_hypograph, ConvexOn.map_average_le, ConcaveOn.le_map_average: assume [NeZero μ] instead of μ ≠ 0;
MeasureTheory.condexp_bot', essSup_const', essInf_const', MeasureTheory.laverage_const, MeasureTheory.laverage_one, MeasureTheory.average_const: assume [NeZero μ] instead of [μ.ae.NeBot]
MeasureTheory.Measure.measure_ne_zero: replace with an instance;
remove @[simp] from MeasureTheory.ae_restrict_neBot, use ≠ 0 in the RHS;
turn MeasureTheory.IsProbabilityMeasure.ae_neBot into a theorem because inferInstance can find it now;
add instances:
- [NeZero μ] : NeZero (μ univ);
- [NeZero (μ s)] : NeZero (μ.restrict s);
- [NeZero μ] : μ.ae.NeBot;
- [IsProbabilityMeasure μ] : NeZero μ;
- [IsFiniteMeasure μ] [NeZero μ] : IsProbabilityMeasure ((μ univ)⁻¹ • μ) this was a theorem MeasureTheory.isProbabilityMeasureSmul assuming μ ≠ 0;

f8c1566c

Diff

@@ -197,11 +197,9 @@ theorem laverage_mem_openSegment_compl_self [IsFiniteMeasure μ] (hs : NullMeasu
 #align measure_theory.laverage_mem_open_segment_compl_self MeasureTheory.laverage_mem_openSegment_compl_self
 
 @[simp]
-theorem laverage_const (μ : Measure α) [IsFiniteMeasure μ] [h : μ.ae.NeBot] (c : ℝ≥0∞) :
+theorem laverage_const (μ : Measure α) [IsFiniteMeasure μ] [h : NeZero μ] (c : ℝ≥0∞) :
     ⨍⁻ _x, c ∂μ = c := by
-  simp only [laverage_eq, lintegral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter,
-    div_eq_mul_inv, mul_assoc, ENNReal.mul_inv_cancel, mul_one, measure_ne_top μ univ, Ne.def,
-    measure_univ_ne_zero, ae_neBot.1 h, not_false_iff]
+  simp only [laverage, lintegral_const, measure_univ, mul_one]
 #align measure_theory.laverage_const MeasureTheory.laverage_const
 
 theorem setLaverage_const (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : ℝ≥0∞) : ⨍⁻ _x in s, c ∂μ = c := by
@@ -209,7 +207,7 @@ theorem setLaverage_const (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : ℝ≥0
     univ_inter, div_eq_mul_inv, mul_assoc, ENNReal.mul_inv_cancel hs₀ hs, mul_one]
 #align measure_theory.set_laverage_const MeasureTheory.setLaverage_const
 
-theorem laverage_one [IsFiniteMeasure μ] [μ.ae.NeBot] : ⨍⁻ _x, (1 : ℝ≥0∞) ∂μ = 1 :=
+theorem laverage_one [IsFiniteMeasure μ] [NeZero μ] : ⨍⁻ _x, (1 : ℝ≥0∞) ∂μ = 1 :=
   laverage_const _ _
 #align measure_theory.laverage_one MeasureTheory.laverage_one
 
@@ -387,19 +385,14 @@ theorem average_mem_openSegment_compl_self [IsFiniteMeasure μ] {f : α → E} {
 #align measure_theory.average_mem_open_segment_compl_self MeasureTheory.average_mem_openSegment_compl_self
 
 @[simp]
-theorem average_const (μ : Measure α) [IsFiniteMeasure μ] [h : μ.ae.NeBot] (c : E) :
+theorem average_const (μ : Measure α) [IsFiniteMeasure μ] [h : NeZero μ] (c : E) :
     ⨍ _x, c ∂μ = c := by
-  simp only [average_eq, integral_const, Measure.restrict_apply, MeasurableSet.univ, one_smul,
-    univ_inter, smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel,
-    measure_ne_top μ univ, Ne.def, measure_univ_eq_zero, ae_neBot.1 h, not_false_iff,
-    ENNReal.one_toReal]
+  rw [average, integral_const, measure_univ, ENNReal.one_toReal, one_smul]
 #align measure_theory.average_const MeasureTheory.average_const
 
 theorem setAverage_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : E) :
-    ⨍ _ in s, c ∂μ = c := by
-  simp only [setAverage_eq, integral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter,
-    smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel hs₀ hs,
-    ENNReal.one_toReal, one_smul]
+    ⨍ _ in s, c ∂μ = c :=
+  have := NeZero.mk hs₀; have := Fact.mk hs.lt_top; average_const _ _
 #align measure_theory.set_average_const MeasureTheory.setAverage_const
 
 -- porting note: was `@[simp]` but `simp` can prove it

chore: script to replace headers with #align_import statements (#5979)

depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

89aa3a3b

Diff

@@ -2,14 +2,11 @@
 Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov, Yaël Dillies
-
-! This file was ported from Lean 3 source module measure_theory.integral.average
-! leanprover-community/mathlib commit c14c8fcde993801fca8946b0d80131a1a81d1520
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.MeasureTheory.Integral.SetIntegral
 
+#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
+
 /-!
 # Integral average of a function

Match https://github.com/leanprover-community/mathlib/pull/19199

feat: Lebesgue average (#5810)

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

44d1fd93

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov, Yaël Dillies
 
 ! This file was ported from Lean 3 source module measure_theory.integral.average
-! leanprover-community/mathlib commit ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f
+! leanprover-community/mathlib commit c14c8fcde993801fca8946b0d80131a1a81d1520
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -21,6 +21,8 @@ measure, then the average of any function is equal to its integral.
 For the average on a set, we use `⨍ x in s, f x ∂μ` (notation for `⨍ x, f x ∂(μ.restrict s)`). For
 average w.r.t. the volume, one can omit `∂volume`.
 
+Both have a version for the Lebesgue integral rather than Bochner.
+
 We prove several version of the first moment method: An integrable function is below/above its
 average on a set of positive measure.
 
@@ -46,20 +48,194 @@ open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSp
 open scoped Topology BigOperators ENNReal Convex
 
 variable {α E F : Type _} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {μ : Measure α}
+  [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {μ ν : Measure α}
   {s t : Set α}
 
 /-!
 ### Average value of a function w.r.t. a measure
 
-The average value of a function `f` w.r.t. a measure `μ` (notation: `⨍ x, f x ∂μ`) is defined as
-`(μ univ).toReal⁻¹ • ∫ x, f x ∂μ`, so it is equal to zero if `f` is not integrable or if `μ` is an
-infinite measure. If `μ` is a probability measure, then the average of any function is equal to its
-integral.
-
+The (Bochner, Lebesgue) average value of a function `f` w.r.t. a measure `μ` (notation:
+`⨍ x, f x ∂μ`, `⨍⁻ x, f x ∂μ`) is defined as the (Bochner, Lebesgue) integral divided by the total
+measure, so it is equal to zero if `μ` is an infinite measure, and (typically) equal to infinity if
+`f` is not integrable. If `μ` is a probability measure, then the average of any function is equal to
+its integral.
 -/
 
 namespace MeasureTheory
+section ENNReal
+variable (μ) {f g : α → ℝ≥0∞}
+
+/-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. a measure `μ`, notation: `⨍⁻ x, f x ∂μ`.
+It is defined as `μ univ⁻¹ * ∫⁻ x, f x ∂μ`, so it is equal to zero if `μ` is an infinite measure. If
+`μ` is a probability measure, then the average of any function is equal to its integral.
+
+For the average on a set, use `⨍⁻ x in s, f x ∂μ` (defined as `⨍⁻ x, f x ∂(μ.restrict s)`). For
+average w.r.t. the volume, one can omit `∂volume`. -/
+noncomputable def laverage (f : α → ℝ≥0∞) := ∫⁻ x, f x ∂(μ univ)⁻¹ • μ
+#align measure_theory.laverage MeasureTheory.laverage
+
+notation3"⨍⁻ "(...)", "r:60:(scoped f => f)" ∂"μ:70 => laverage μ r
+notation3"⨍⁻ "(...)", "r:60:(scoped f => laverage volume f) => r
+notation3"⨍⁻ "(...)" in "s", "r:60:(scoped f => f)" ∂"μ:70 => laverage (Measure.restrict μ s) r
+notation3 (prettyPrint := false)
+  "⨍⁻ "(...)" in "s", "r:60:(scoped f => laverage Measure.restrict volume s f) => r
+
+@[simp]
+theorem laverage_zero : ⨍⁻ _x, (0 : ℝ≥0∞) ∂μ = 0 := by rw [laverage, lintegral_zero]
+#align measure_theory.laverage_zero MeasureTheory.laverage_zero
+
+@[simp]
+theorem laverage_zero_measure (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂(0 : Measure α) = 0 := by simp [laverage]
+#align measure_theory.laverage_zero_measure MeasureTheory.laverage_zero_measure
+
+theorem laverage_eq' (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂(μ univ)⁻¹ • μ := rfl
+#align measure_theory.laverage_eq' MeasureTheory.laverage_eq'
+
+theorem laverage_eq (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = (∫⁻ x, f x ∂μ) / μ univ := by
+  rw [laverage_eq', lintegral_smul_measure, ENNReal.div_eq_inv_mul]
+#align measure_theory.laverage_eq MeasureTheory.laverage_eq
+
+theorem laverage_eq_lintegral [IsProbabilityMeasure μ] (f : α → ℝ≥0∞) :
+    ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ := by rw [laverage, measure_univ, inv_one, one_smul]
+#align measure_theory.laverage_eq_lintegral MeasureTheory.laverage_eq_lintegral
+
+@[simp]
+theorem measure_mul_laverage [IsFiniteMeasure μ] (f : α → ℝ≥0∞) :
+    μ univ * ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ := by
+  cases' eq_or_ne μ 0 with hμ hμ
+  · rw [hμ, lintegral_zero_measure, laverage_zero_measure, MulZeroClass.mul_zero]
+  · rw [laverage_eq, ENNReal.mul_div_cancel' (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)]
+#align measure_theory.measure_mul_laverage MeasureTheory.measure_mul_laverage
+
+theorem setLaverage_eq (f : α → ℝ≥0∞) (s : Set α) :
+    ⨍⁻ x in s, f x ∂μ = (∫⁻ x in s, f x ∂μ) / μ s := by rw [laverage_eq, restrict_apply_univ]
+#align measure_theory.set_laverage_eq MeasureTheory.setLaverage_eq
+
+theorem setLaverage_eq' (f : α → ℝ≥0∞) (s : Set α) :
+    ⨍⁻ x in s, f x ∂μ = ∫⁻ x, f x ∂(μ s)⁻¹ • μ.restrict s := by
+  simp only [laverage_eq', restrict_apply_univ]
+#align measure_theory.set_laverage_eq' MeasureTheory.setLaverage_eq'
+
+variable {μ}
+
+theorem laverage_congr {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : ⨍⁻ x, f x ∂μ = ⨍⁻ x, g x ∂μ := by
+  simp only [laverage_eq, lintegral_congr_ae h]
+#align measure_theory.laverage_congr MeasureTheory.laverage_congr
+
+theorem setLaverage_congr (h : s =ᵐ[μ] t) : ⨍⁻ x in s, f x ∂μ = ⨍⁻ x in t, f x ∂μ := by
+  simp only [setLaverage_eq, set_lintegral_congr h, measure_congr h]
+#align measure_theory.set_laverage_congr MeasureTheory.setLaverage_congr
+
+theorem setLaverage_congr_fun (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
+    ⨍⁻ x in s, f x ∂μ = ⨍⁻ x in s, g x ∂μ := by
+  simp only [laverage_eq, set_lintegral_congr_fun hs h]
+#align measure_theory.set_laverage_congr_fun MeasureTheory.setLaverage_congr_fun
+
+theorem laverage_lt_top (hf : ∫⁻ x, f x ∂μ ≠ ∞) : ⨍⁻ x, f x ∂μ < ∞ := by
+  obtain rfl | hμ := eq_or_ne μ 0
+  · simp
+  · rw [laverage_eq]
+    exact div_lt_top hf (measure_univ_ne_zero.2 hμ)
+#align measure_theory.laverage_lt_top MeasureTheory.laverage_lt_top
+
+theorem setLaverage_lt_top : ∫⁻ x in s, f x ∂μ ≠ ∞ → ⨍⁻ x in s, f x ∂μ < ∞ :=
+  laverage_lt_top
+#align measure_theory.set_laverage_lt_top MeasureTheory.setLaverage_lt_top
+
+theorem laverage_add_measure :
+    ⨍⁻ x, f x ∂(μ + ν) =
+      μ univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂μ + ν univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂ν := by
+  by_cases hμ : IsFiniteMeasure μ; swap
+  · rw [not_isFiniteMeasure_iff] at hμ
+    simp [laverage_eq, hμ]
+  by_cases hν : IsFiniteMeasure ν; swap
+  · rw [not_isFiniteMeasure_iff] at hν
+    simp [laverage_eq, hν]
+  haveI := hμ; haveI := hν
+  simp only [←ENNReal.mul_div_right_comm, measure_mul_laverage, ←ENNReal.add_div,
+    ←lintegral_add_measure, ←Measure.add_apply, ←laverage_eq]
+#align measure_theory.laverage_add_measure MeasureTheory.laverage_add_measure
+
+theorem measure_mul_setLaverage (f : α → ℝ≥0∞) (h : μ s ≠ ∞) :
+    μ s * ⨍⁻ x in s, f x ∂μ = ∫⁻ x in s, f x ∂μ := by
+  have := Fact.mk h.lt_top
+  rw [←measure_mul_laverage, restrict_apply_univ]
+#align measure_theory.measure_mul_set_laverage MeasureTheory.measure_mul_setLaverage
+
+theorem laverage_union (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ) :
+    ⨍⁻ x in s ∪ t, f x ∂μ =
+      μ s / (μ s + μ t) * ⨍⁻ x in s, f x ∂μ + μ t / (μ s + μ t) * ⨍⁻ x in t, f x ∂μ :=
+  by rw [restrict_union₀ hd ht, laverage_add_measure, restrict_apply_univ, restrict_apply_univ]
+#align measure_theory.laverage_union MeasureTheory.laverage_union
+
+theorem laverage_union_mem_openSegment (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
+    (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) :
+    ⨍⁻ x in s ∪ t, f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ x in s, f x ∂μ) (⨍⁻ x in t, f x ∂μ) := by
+  refine'
+    ⟨μ s / (μ s + μ t), μ t / (μ s + μ t), ENNReal.div_pos hs₀ <| add_ne_top.2 ⟨hsμ, htμ⟩,
+      ENNReal.div_pos ht₀ <| add_ne_top.2 ⟨hsμ, htμ⟩, _, (laverage_union hd ht).symm⟩
+  rw [←ENNReal.add_div,
+    ENNReal.div_self (add_eq_zero.not.2 fun h => hs₀ h.1) (add_ne_top.2 ⟨hsμ, htμ⟩)]
+#align measure_theory.laverage_union_mem_open_segment MeasureTheory.laverage_union_mem_openSegment
+
+theorem laverage_union_mem_segment (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
+    (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) :
+    ⨍⁻ x in s ∪ t, f x ∂μ ∈ [⨍⁻ x in s, f x ∂μ -[ℝ≥0∞] ⨍⁻ x in t, f x ∂μ] := by
+  by_cases hs₀ : μ s = 0
+  · rw [←ae_eq_empty] at hs₀
+    rw [restrict_congr_set (hs₀.union EventuallyEq.rfl), empty_union]
+    exact right_mem_segment _ _ _
+  · refine'
+      ⟨μ s / (μ s + μ t), μ t / (μ s + μ t), zero_le _, zero_le _, _, (laverage_union hd ht).symm⟩
+    rw [←ENNReal.add_div,
+      ENNReal.div_self (add_eq_zero.not.2 fun h => hs₀ h.1) (add_ne_top.2 ⟨hsμ, htμ⟩)]
+#align measure_theory.laverage_union_mem_segment MeasureTheory.laverage_union_mem_segment
+
+theorem laverage_mem_openSegment_compl_self [IsFiniteMeasure μ] (hs : NullMeasurableSet s μ)
+    (hs₀ : μ s ≠ 0) (hsc₀ : μ sᶜ ≠ 0) :
+    ⨍⁻ x, f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ x in s, f x ∂μ) (⨍⁻ x in sᶜ, f x ∂μ) := by
+  simpa only [union_compl_self, restrict_univ] using
+    laverage_union_mem_openSegment aedisjoint_compl_right hs.compl hs₀ hsc₀ (measure_ne_top _ _)
+      (measure_ne_top _ _)
+#align measure_theory.laverage_mem_open_segment_compl_self MeasureTheory.laverage_mem_openSegment_compl_self
+
+@[simp]
+theorem laverage_const (μ : Measure α) [IsFiniteMeasure μ] [h : μ.ae.NeBot] (c : ℝ≥0∞) :
+    ⨍⁻ _x, c ∂μ = c := by
+  simp only [laverage_eq, lintegral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter,
+    div_eq_mul_inv, mul_assoc, ENNReal.mul_inv_cancel, mul_one, measure_ne_top μ univ, Ne.def,
+    measure_univ_ne_zero, ae_neBot.1 h, not_false_iff]
+#align measure_theory.laverage_const MeasureTheory.laverage_const
+
+theorem setLaverage_const (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : ℝ≥0∞) : ⨍⁻ _x in s, c ∂μ = c := by
+  simp only [setLaverage_eq, lintegral_const, Measure.restrict_apply, MeasurableSet.univ,
+    univ_inter, div_eq_mul_inv, mul_assoc, ENNReal.mul_inv_cancel hs₀ hs, mul_one]
+#align measure_theory.set_laverage_const MeasureTheory.setLaverage_const
+
+theorem laverage_one [IsFiniteMeasure μ] [μ.ae.NeBot] : ⨍⁻ _x, (1 : ℝ≥0∞) ∂μ = 1 :=
+  laverage_const _ _
+#align measure_theory.laverage_one MeasureTheory.laverage_one
+
+theorem setLaverage_one (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) : ⨍⁻ _x in s, (1 : ℝ≥0∞) ∂μ = 1 :=
+  setLaverage_const hs₀ hs _
+#align measure_theory.set_laverage_one MeasureTheory.setLaverage_one
+
+--porting note: Dropped `simp` because of `simp` seeing through `1 : α → ℝ≥0∞` and applying
+-- `lintegral_const`. This is suboptimal.
+theorem lintegral_laverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ≥0∞) :
+    ∫⁻ _x, ⨍⁻ a, f a ∂μ ∂μ = ∫⁻ x, f x ∂μ := by
+  obtain rfl | hμ := eq_or_ne μ 0
+  · simp
+  · rw [lintegral_const, laverage_eq,
+      ENNReal.div_mul_cancel (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)]
+#align measure_theory.lintegral_laverage MeasureTheory.lintegral_laverage
+
+theorem setLintegral_setLaverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ≥0∞) (s : Set α) :
+    ∫⁻ _x in s, ⨍⁻ a in s, f a ∂μ ∂μ = ∫⁻ x in s, f x ∂μ :=
+  lintegral_laverage _ _
+#align measure_theory.set_lintegral_set_laverage MeasureTheory.setLintegral_setLaverage
+
+end ENNReal
 
 section NormedAddCommGroup
 
@@ -125,14 +301,14 @@ theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
     rwa [Ne.def, measure_univ_eq_zero]
 #align measure_theory.measure_smul_average MeasureTheory.measure_smul_average
 
-theorem set_average_eq (f : α → E) (s : Set α) :
+theorem setAverage_eq (f : α → E) (s : Set α) :
     ⨍ x in s, f x ∂μ = (μ s).toReal⁻¹ • ∫ x in s, f x ∂μ := by rw [average_eq, restrict_apply_univ]
-#align measure_theory.set_average_eq MeasureTheory.set_average_eq
+#align measure_theory.set_average_eq MeasureTheory.setAverage_eq
 
-theorem set_average_eq' (f : α → E) (s : Set α) :
+theorem setAverage_eq' (f : α → E) (s : Set α) :
     ⨍ x in s, f x ∂μ = ∫ x, f x ∂(μ s)⁻¹ • μ.restrict s := by
   simp only [average_eq', restrict_apply_univ]
-#align measure_theory.set_average_eq' MeasureTheory.set_average_eq'
+#align measure_theory.set_average_eq' MeasureTheory.setAverage_eq'
 
 variable {μ}
 
@@ -140,17 +316,21 @@ theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : ⨍ x, f x ∂μ = 
   simp only [average_eq, integral_congr_ae h]
 #align measure_theory.average_congr MeasureTheory.average_congr
 
-theorem set_average_congr_set_ae (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ := by
-  simp only [set_average_eq, set_integral_congr_set_ae h, measure_congr h]
-#align measure_theory.set_average_congr_set_ae MeasureTheory.set_average_congr_set_ae
+theorem setAverage_congr (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ := by
+  simp only [setAverage_eq, set_integral_congr_set_ae h, measure_congr h]
+#align measure_theory.set_average_congr MeasureTheory.setAverage_congr
+
+theorem setAverage_congr_fun (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
+    ⨍ x in s, f x ∂μ = ⨍ x in s, g x ∂μ := by simp only [average_eq, set_integral_congr_ae hs h]
+#align measure_theory.set_average_congr_fun MeasureTheory.setAverage_congr_fun
 
 theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E}
     (hμ : Integrable f μ) (hν : Integrable f ν) :
     ⨍ x, f x ∂(μ + ν) =
       ((μ univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂μ +
         ((ν univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂ν := by
-  simp only [div_eq_inv_mul, mul_smul, measure_smul_average, ← smul_add, ←
-    integral_add_measure hμ hν, ← ENNReal.toReal_add (measure_ne_top μ _) (measure_ne_top ν _)]
+  simp only [div_eq_inv_mul, mul_smul, measure_smul_average, ←smul_add,
+    ←integral_add_measure hμ hν, ←ENNReal.toReal_add (measure_ne_top μ _) (measure_ne_top ν _)]
   rw [average_eq, Measure.add_apply]
 #align measure_theory.average_add_measure MeasureTheory.average_add_measure
 
@@ -159,11 +339,11 @@ theorem average_pair {f : α → E} {g : α → F} (hfi : Integrable f μ) (hgi
   integral_pair hfi.to_average hgi.to_average
 #align measure_theory.average_pair MeasureTheory.average_pair
 
-theorem measure_smul_set_average (f : α → E) {s : Set α} (h : μ s ≠ ∞) :
+theorem measure_smul_setAverage (f : α → E) {s : Set α} (h : μ s ≠ ∞) :
     (μ s).toReal • ⨍ x in s, f x ∂μ = ∫ x in s, f x ∂μ := by
   haveI := Fact.mk h.lt_top
-  rw [← measure_smul_average, restrict_apply_univ]
-#align measure_theory.measure_smul_set_average MeasureTheory.measure_smul_set_average
+  rw [←measure_smul_average, restrict_apply_univ]
+#align measure_theory.measure_smul_set_average MeasureTheory.measure_smul_setAverage
 
 theorem average_union {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
     (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
@@ -189,7 +369,7 @@ theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint
     (hft : IntegrableOn f t μ) :
     ⨍ x in s ∪ t, f x ∂μ ∈ [⨍ x in s, f x ∂μ -[ℝ] ⨍ x in t, f x ∂μ] := by
   by_cases hse : μ s = 0
-  · rw [← ae_eq_empty] at hse
+  · rw [←ae_eq_empty] at hse
     rw [restrict_congr_set (hse.union EventuallyEq.rfl), empty_union]
     exact right_mem_segment _ _ _
   · refine'
@@ -210,29 +390,30 @@ theorem average_mem_openSegment_compl_self [IsFiniteMeasure μ] {f : α → E} {
 #align measure_theory.average_mem_open_segment_compl_self MeasureTheory.average_mem_openSegment_compl_self
 
 @[simp]
-theorem average_const [IsFiniteMeasure μ] [h : μ.ae.NeBot] (c : E) : ⨍ _, c ∂μ = c := by
+theorem average_const (μ : Measure α) [IsFiniteMeasure μ] [h : μ.ae.NeBot] (c : E) :
+    ⨍ _x, c ∂μ = c := by
   simp only [average_eq, integral_const, Measure.restrict_apply, MeasurableSet.univ, one_smul,
     univ_inter, smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel,
     measure_ne_top μ univ, Ne.def, measure_univ_eq_zero, ae_neBot.1 h, not_false_iff,
     ENNReal.one_toReal]
 #align measure_theory.average_const MeasureTheory.average_const
 
-theorem set_average_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : E) :
+theorem setAverage_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : E) :
     ⨍ _ in s, c ∂μ = c := by
-  simp only [set_average_eq, integral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter,
+  simp only [setAverage_eq, integral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter,
     smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel hs₀ hs,
     ENNReal.one_toReal, one_smul]
-#align measure_theory.set_average_const MeasureTheory.set_average_const
+#align measure_theory.set_average_const MeasureTheory.setAverage_const
 
 -- porting note: was `@[simp]` but `simp` can prove it
 theorem integral_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :
     ∫ _, ⨍ a, f a ∂μ ∂μ = ∫ x, f x ∂μ := by simp
 #align measure_theory.integral_average MeasureTheory.integral_average
 
-theorem setAverage_setAverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) (s : Set α) :
+theorem setIntegral_setAverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) (s : Set α) :
     ∫ _ in s, ⨍ a in s, f a ∂μ ∂μ = ∫ x in s, f x ∂μ :=
   integral_average _ _
-#align measure_theory.set_integral_set_average MeasureTheory.setAverage_setAverage
+#align measure_theory.set_integral_set_average MeasureTheory.setIntegral_setAverage
 
 theorem integral_sub_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :
     ∫ x, f x - ⨍ a, f a ∂μ ∂μ = 0 := by
@@ -254,11 +435,11 @@ theorem integral_average_sub [IsFiniteMeasure μ] (hf : Integrable f μ) :
   rw [integral_sub (integrable_const _) hf, integral_average, sub_self]
 #align measure_theory.integral_average_sub MeasureTheory.integral_average_sub
 
-theorem setAverage_setAverage_sub (hs : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+theorem setIntegral_setAverage_sub (hs : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
     ∫ x in s, ⨍ a in s, f a ∂μ - f x ∂μ = 0 :=
   haveI : Fact (μ s < ∞) := ⟨lt_top_iff_ne_top.2 hs⟩
   integral_average_sub hf
-#align measure_theory.set_integral_set_average_sub MeasureTheory.setAverage_setAverage_sub
+#align measure_theory.set_integral_set_average_sub MeasureTheory.setIntegral_setAverage_sub
 
 end NormedAddCommGroup
 
@@ -276,22 +457,21 @@ theorem ofReal_setAverage {f : α → ℝ} (hf : IntegrableOn f s μ) (hf₀ : 0
   simpa using ofReal_average hf hf₀
 #align measure_theory.of_real_set_average MeasureTheory.ofReal_setAverage
 
-theorem average_toReal {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf' : ∀ᵐ x ∂μ, f x ≠ ∞) :
-    ⨍ x, (f x).toReal ∂μ = ((∫⁻ x, f x ∂μ) / μ univ).toReal := by
-  rw [average_eq, smul_eq_mul, toReal_div, ←
-    integral_toReal hf (hf'.mono fun _ => lt_top_iff_ne_top.2), div_eq_inv_mul]
-#align measure_theory.average_to_real MeasureTheory.average_toReal
+theorem toReal_laverage {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf' : ∀ᵐ x ∂μ, f x ≠ ∞) :
+    (⨍⁻ x, f x ∂μ).toReal = ⨍ x, (f x).toReal ∂μ := by
+    rw [average_eq, laverage_eq, smul_eq_mul, toReal_div, div_eq_inv_mul, ←
+      integral_toReal hf (hf'.mono fun _ => lt_top_iff_ne_top.2)]
+#align measure_theory.to_real_laverage MeasureTheory.toReal_laverage
 
-theorem setAverage_toReal {f : α → ℝ≥0∞} (hf : AEMeasurable f (μ.restrict s))
+theorem toReal_setLaverage {f : α → ℝ≥0∞} (hf : AEMeasurable f (μ.restrict s))
     (hf' : ∀ᵐ x ∂μ.restrict s, f x ≠ ∞) :
-    ⨍ x in s, (f x).toReal ∂μ = ((∫⁻ x in s, f x ∂μ) / μ s).toReal := by
-  simpa using average_toReal hf hf'
-#align measure_theory.set_average_to_real MeasureTheory.setAverage_toReal
+    (⨍⁻ x in s, f x ∂μ).toReal = ⨍ x in s, (f x).toReal ∂μ := by
+  simpa [laverage_eq] using toReal_laverage hf hf'
+#align measure_theory.to_real_set_laverage MeasureTheory.toReal_setLaverage
 
 /-! ### First moment method -/
 
-section FirstMoment
-
+section FirstMomentReal
 variable {N : Set α} {f : α → ℝ}
 
 /-- **First moment method**. An integrable function is smaller than its mean on a set of positive
@@ -431,7 +611,166 @@ theorem exists_not_mem_null_integral_le (hf : Integrable f μ) (hN : μ N = 0) :
 #align measure_theory.exists_not_mem_null_integral_le MeasureTheory.exists_not_mem_null_integral_le
 
 end ProbabilityMeasure
+end FirstMomentReal
+
+section FirstMomentEnnreal
+variable {N : Set α} {f : α → ℝ≥0∞}
+
+/-- **First moment method**. A measurable function is smaller than its mean on a set of ositive
+measure. -/
+theorem measure_le_setLaverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞)
+    (hf : AEMeasurable f (μ.restrict s)) : 0 < μ {x ∈ s | f x ≤ ⨍⁻ a in s, f a ∂μ} := by
+  obtain h | h := eq_or_ne (∫⁻ a in s, f a ∂μ) ∞
+  · simpa [mul_top, hμ₁, laverage, h, top_div_of_ne_top hμ₁, pos_iff_ne_zero] using hμ
+  have := measure_le_setAverage_pos hμ hμ₁ (integrable_toReal_of_lintegral_ne_top hf h)
+  rw [←setOf_inter_eq_sep, ←Measure.restrict_apply₀
+    (hf.aestronglyMeasurable.nullMeasurableSet_le aestronglyMeasurable_const)]
+  rw [←setOf_inter_eq_sep, ←Measure.restrict_apply₀
+    (hf.ennreal_toReal.aestronglyMeasurable.nullMeasurableSet_le aestronglyMeasurable_const),
+    ←measure_diff_null (measure_eq_top_of_lintegral_ne_top hf h)] at this
+  refine' this.trans_le (measure_mono _)
+  rintro x ⟨hfx, hx⟩
+  dsimp at hfx
+  rwa [←toReal_laverage hf, toReal_le_toReal hx (setLaverage_lt_top h).ne] at hfx
+  · simp_rw [ae_iff, not_ne_iff]
+    exact measure_eq_top_of_lintegral_ne_top hf h
+#align measure_theory.measure_le_set_laverage_pos MeasureTheory.measure_le_setLaverage_pos
+
+/-- **First moment method**. A measurable function is greater than its mean on a set of ositive
+measure. -/
+theorem measure_setLaverage_le_pos (hμ : μ s ≠ 0) (hs : NullMeasurableSet s μ)
+    (hint : ∫⁻ a in s, f a ∂μ ≠ ∞) : 0 < μ {x ∈ s | ⨍⁻ a in s, f a ∂μ ≤ f x} := by
+  obtain hμ₁ | hμ₁ := eq_or_ne (μ s) ∞
+  · simp [setLaverage_eq, hμ₁]
+  obtain ⟨g, hg, hgf, hfg⟩ := exists_measurable_le_lintegral_eq (μ.restrict s) f
+  have hfg' : ⨍⁻ a in s, f a ∂μ = ⨍⁻ a in s, g a ∂μ := by simp_rw [laverage_eq, hfg]
+  rw [hfg] at hint
+  have :=
+    measure_setAverage_le_pos hμ hμ₁ (integrable_toReal_of_lintegral_ne_top hg.aemeasurable hint)
+  simp_rw [←setOf_inter_eq_sep, ←Measure.restrict_apply₀' hs, hfg']
+  rw [←setOf_inter_eq_sep, ←Measure.restrict_apply₀' hs, ←
+    measure_diff_null (measure_eq_top_of_lintegral_ne_top hg.aemeasurable hint)] at this
+  refine' this.trans_le (measure_mono _)
+  rintro x ⟨hfx, hx⟩
+  dsimp at hfx
+  rw [←toReal_laverage hg.aemeasurable, toReal_le_toReal (setLaverage_lt_top hint).ne hx] at hfx
+  exact hfx.trans (hgf _)
+  · simp_rw [ae_iff, not_ne_iff]
+    exact measure_eq_top_of_lintegral_ne_top hg.aemeasurable hint
+#align measure_theory.measure_set_laverage_le_pos MeasureTheory.measure_setLaverage_le_pos
+
+/-- **First moment method**. The minimum of a measurable function is smaller than its ean. -/
+theorem exists_le_setLaverage (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : AEMeasurable f (μ.restrict s)) :
+    ∃ x ∈ s, f x ≤ ⨍⁻ a in s, f a ∂μ :=
+  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_le_setLaverage_pos hμ hμ₁ hf).ne'
+  ⟨x, hx, h⟩
+#align measure_theory.exists_le_set_laverage MeasureTheory.exists_le_setLaverage
+
+/-- **First moment method**. The maximum of a measurable function is greater than its ean. -/
+theorem exists_setLaverage_le (hμ : μ s ≠ 0) (hs : NullMeasurableSet s μ)
+    (hint : ∫⁻ a in s, f a ∂μ ≠ ∞) : ∃ x ∈ s, ⨍⁻ a in s, f a ∂μ ≤ f x :=
+  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_setLaverage_le_pos hμ hs hint).ne'
+  ⟨x, hx, h⟩
+#align measure_theory.exists_set_laverage_le MeasureTheory.exists_setLaverage_le
+
+/-- **First moment method**. A measurable function is greater than its mean on a set of ositive
+measure. -/
+theorem measure_laverage_le_pos (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞) :
+    0 < μ {x | ⨍⁻ a, f a ∂μ ≤ f x} := by
+  simpa [hint] using
+    @measure_setLaverage_le_pos _ _ _ _ f (measure_univ_ne_zero.2 hμ) nullMeasurableSet_univ
+#align measure_theory.measure_laverage_le_pos MeasureTheory.measure_laverage_le_pos
+
+/-- **First moment method**. The maximum of a measurable function is greater than its ean. -/
+theorem exists_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞) : ∃ x, ⨍⁻ a, f a ∂μ ≤ f x :=
+  let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_laverage_le_pos hμ hint).ne'
+  ⟨x, hx⟩
+#align measure_theory.exists_laverage_le MeasureTheory.exists_laverage_le
+
+/-- **First moment method**. The maximum of a measurable function is greater than its ean, while
+avoiding a null set. -/
+theorem exists_not_mem_null_laverage_le (hμ : μ ≠ 0) (hint : ∫⁻ a : α, f a ∂μ ≠ ∞) (hN : μ N = 0) :
+    ∃ x, x ∉ N ∧ ⨍⁻ a, f a ∂μ ≤ f x := by
+  have := measure_laverage_le_pos hμ hint
+  rw [←measure_diff_null hN] at this
+  obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
+  exact ⟨x, hxN, hx⟩
+#align measure_theory.exists_not_mem_null_laverage_le MeasureTheory.exists_not_mem_null_laverage_le
+
+section FiniteMeasure
+variable [IsFiniteMeasure μ]
+
+/-- **First moment method**. A measurable function is smaller than its mean on a set of ositive
+measure. -/
+theorem measure_le_laverage_pos (hμ : μ ≠ 0) (hf : AEMeasurable f μ) :
+    0 < μ {x | f x ≤ ⨍⁻ a, f a ∂μ} := by
+  simpa using
+    measure_le_setLaverage_pos (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _) hf.restrict
+#align measure_theory.measure_le_laverage_pos MeasureTheory.measure_le_laverage_pos
+
+/-- **First moment method**. The minimum of a measurable function is smaller than its ean. -/
+theorem exists_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) : ∃ x, f x ≤ ⨍⁻ a, f a ∂μ :=
+  let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_le_laverage_pos hμ hf).ne'
+  ⟨x, hx⟩
+#align measure_theory.exists_le_laverage MeasureTheory.exists_le_laverage
+
+/-- **First moment method**. The minimum of a measurable function is smaller than its ean, while
+avoiding a null set. -/
+theorem exists_not_mem_null_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) (hN : μ N = 0) :
+    ∃ x, x ∉ N ∧ f x ≤ ⨍⁻ a, f a ∂μ := by
+  have := measure_le_laverage_pos hμ hf
+  rw [←measure_diff_null hN] at this
+  obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
+  exact ⟨x, hxN, hx⟩
+#align measure_theory.exists_not_mem_null_le_laverage MeasureTheory.exists_not_mem_null_le_laverage
 
-end FirstMoment
+end FiniteMeasure
 
+section ProbabilityMeasure
+
+variable [IsProbabilityMeasure μ]
+
+/-- **First moment method**. A measurable function is smaller than its integral on a set f
+positive measure. -/
+theorem measure_le_lintegral_pos (hf : AEMeasurable f μ) : 0 < μ {x | f x ≤ ∫⁻ a, f a ∂μ} := by
+  simpa only [laverage_eq_lintegral] using
+    measure_le_laverage_pos (IsProbabilityMeasure.ne_zero μ) hf
+#align measure_theory.measure_le_lintegral_pos MeasureTheory.measure_le_lintegral_pos
+
+/-- **First moment method**. A measurable function is greater than its integral on a set f
+positive measure. -/
+theorem measure_lintegral_le_pos (hint : ∫⁻ a, f a ∂μ ≠ ∞) : 0 < μ {x | ∫⁻ a, f a ∂μ ≤ f x} := by
+  simpa only [laverage_eq_lintegral] using
+    measure_laverage_le_pos (IsProbabilityMeasure.ne_zero μ) hint
+#align measure_theory.measure_lintegral_le_pos MeasureTheory.measure_lintegral_le_pos
+
+/-- **First moment method**. The minimum of a measurable function is smaller than its ntegral. -/
+theorem exists_le_lintegral (hf : AEMeasurable f μ) : ∃ x, f x ≤ ∫⁻ a, f a ∂μ := by
+  simpa only [laverage_eq_lintegral] using exists_le_laverage (IsProbabilityMeasure.ne_zero μ) hf
+#align measure_theory.exists_le_lintegral MeasureTheory.exists_le_lintegral
+
+/-- **First moment method**. The maximum of a measurable function is greater than its ntegral. -/
+theorem exists_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) : ∃ x, ∫⁻ a, f a ∂μ ≤ f x := by
+  simpa only [laverage_eq_lintegral] using
+    exists_laverage_le (IsProbabilityMeasure.ne_zero μ) hint
+#align measure_theory.exists_lintegral_le MeasureTheory.exists_lintegral_le
+
+/-- **First moment method**. The minimum of a measurable function is smaller than its ntegral,
+while avoiding a null set. -/
+theorem exists_not_mem_null_le_lintegral (hf : AEMeasurable f μ) (hN : μ N = 0) :
+    ∃ x, x ∉ N ∧ f x ≤ ∫⁻ a, f a ∂μ := by
+  simpa only [laverage_eq_lintegral] using
+    exists_not_mem_null_le_laverage (IsProbabilityMeasure.ne_zero μ) hf hN
+#align measure_theory.exists_not_mem_null_le_lintegral MeasureTheory.exists_not_mem_null_le_lintegral
+
+/-- **First moment method**. The maximum of a measurable function is greater than its ntegral,
+while avoiding a null set. -/
+theorem exists_not_mem_null_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) (hN : μ N = 0) :
+    ∃ x, x ∉ N ∧ ∫⁻ a, f a ∂μ ≤ f x := by
+  simpa only [laverage_eq_lintegral] using
+    exists_not_mem_null_laverage_le (IsProbabilityMeasure.ne_zero μ) hint hN
+#align measure_theory.exists_not_mem_null_lintegral_le MeasureTheory.exists_not_mem_null_lintegral_le
+
+end ProbabilityMeasure
+end FirstMomentEnnreal
 end MeasureTheory

feat: first moment method (#5326)

Forward-port leanprover-community/mathlib#18731 Also sync some parentheses with the lates mathport output.

Co-authored-by: Yaël Dillies <yael.dillies@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

9e3ff9e1

Diff

@@ -1,10 +1,10 @@
 /-
 Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Yury G. Kudryashov
+Authors: Yury G. Kudryashov, Yaël Dillies
 
 ! This file was ported from Lean 3 source module measure_theory.integral.average
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -21,25 +21,33 @@ measure, then the average of any function is equal to its integral.
 For the average on a set, we use `⨍ x in s, f x ∂μ` (notation for `⨍ x, f x ∂(μ.restrict s)`). For
 average w.r.t. the volume, one can omit `∂volume`.
 
+We prove several version of the first moment method: An integrable function is below/above its
+average on a set of positive measure.
+
 ## Implementation notes
 
 The average is defined as an integral over `(μ univ)⁻¹ • μ` so that all theorems about Bochner
 integrals work for the average without modifications. For theorems that require integrability of a
 function, we provide a convenience lemma `MeasureTheory.Integrable.to_average`.
 
+## TODO
+
+Provide the first moment method for the Lebesgue integral as well. A draft is available on branch
+`first_moment_lintegral` in mathlib3 repository.
+
 ## Tags
 
 integral, center mass, average value
 -/
 
 
-open MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
+open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
 
 open scoped Topology BigOperators ENNReal Convex
 
 variable {α E F : Type _} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E]
   [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {μ : Measure α}
-  {s : Set E}
+  {s t : Set α}
 
 /-!
 ### Average value of a function w.r.t. a measure
@@ -51,10 +59,12 @@ integral.
 
 -/
 
-
 namespace MeasureTheory
 
+section NormedAddCommGroup
+
 variable (μ)
+variable {f g : α → E}
 
 /-- Average value of a function `f` w.r.t. a measure `μ`, notation: `⨍ x, f x ∂μ`. It is defined as
 `(μ univ).toReal⁻¹ • ∫ x, f x ∂μ`, so it is equal to zero if `f` is not integrable or if `μ` is an
@@ -84,7 +94,7 @@ theorem average_zero : ⨍ _, (0 : E) ∂μ = 0 := by rw [average, integral_zero
 #align measure_theory.average_zero MeasureTheory.average_zero
 
 @[simp]
-theorem average_zero_measure (f : α → E) : (⨍ x, f x ∂(0 : Measure α)) = 0 := by
+theorem average_zero_measure (f : α → E) : ⨍ x, f x ∂(0 : Measure α) = 0 := by
   rw [average, smul_zero, integral_zero_measure]
 #align measure_theory.average_zero_measure MeasureTheory.average_zero_measure
 
@@ -107,7 +117,7 @@ theorem average_eq_integral [IsProbabilityMeasure μ] (f : α → E) : ⨍ x, f
 
 @[simp]
 theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
-    ((μ univ).toReal • ⨍ x, f x ∂μ) = ∫ x, f x ∂μ := by
+    (μ univ).toReal • ⨍ x, f x ∂μ = ∫ x, f x ∂μ := by
   cases' eq_or_ne μ 0 with hμ hμ
   · rw [hμ, integral_zero_measure, average_zero_measure, smul_zero]
   · rw [average_eq, smul_inv_smul₀]
@@ -116,8 +126,7 @@ theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
 #align measure_theory.measure_smul_average MeasureTheory.measure_smul_average
 
 theorem set_average_eq (f : α → E) (s : Set α) :
-    ⨍ x in s, f x ∂μ = (μ s).toReal⁻¹ • ∫ x in s, f x ∂μ := by
-  rw [average_eq, restrict_apply_univ]
+    ⨍ x in s, f x ∂μ = (μ s).toReal⁻¹ • ∫ x in s, f x ∂μ := by rw [average_eq, restrict_apply_univ]
 #align measure_theory.set_average_eq MeasureTheory.set_average_eq
 
 theorem set_average_eq' (f : α → E) (s : Set α) :
@@ -131,10 +140,14 @@ theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : ⨍ x, f x ∂μ = 
   simp only [average_eq, integral_congr_ae h]
 #align measure_theory.average_congr MeasureTheory.average_congr
 
+theorem set_average_congr_set_ae (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ := by
+  simp only [set_average_eq, set_integral_congr_set_ae h, measure_congr h]
+#align measure_theory.set_average_congr_set_ae MeasureTheory.set_average_congr_set_ae
+
 theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E}
     (hμ : Integrable f μ) (hν : Integrable f ν) :
     ⨍ x, f x ∂(μ + ν) =
-      (((μ univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂μ) +
+      ((μ univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂μ +
         ((ν univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂ν := by
   simp only [div_eq_inv_mul, mul_smul, measure_smul_average, ← smul_add, ←
     integral_add_measure hμ hν, ← ENNReal.toReal_add (measure_ne_top μ _) (measure_ne_top ν _)]
@@ -147,7 +160,7 @@ theorem average_pair {f : α → E} {g : α → F} (hfi : Integrable f μ) (hgi
 #align measure_theory.average_pair MeasureTheory.average_pair
 
 theorem measure_smul_set_average (f : α → E) {s : Set α} (h : μ s ≠ ∞) :
-    ((μ s).toReal • ⨍ x in s, f x ∂μ) = ∫ x in s, f x ∂μ := by
+    (μ s).toReal • ⨍ x in s, f x ∂μ = ∫ x in s, f x ∂μ := by
   haveI := Fact.mk h.lt_top
   rw [← measure_smul_average, restrict_apply_univ]
 #align measure_theory.measure_smul_set_average MeasureTheory.measure_smul_set_average
@@ -155,7 +168,7 @@ theorem measure_smul_set_average (f : α → E) {s : Set α} (h : μ s ≠ ∞)
 theorem average_union {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
     (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
     ⨍ x in s ∪ t, f x ∂μ =
-      (((μ s).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ x in s, f x ∂μ) +
+      ((μ s).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ x in s, f x ∂μ +
         ((μ t).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ x in t, f x ∂μ := by
   haveI := Fact.mk hsμ.lt_top; haveI := Fact.mk htμ.lt_top
   rw [restrict_union₀ hd ht, average_add_measure hfs hft, restrict_apply_univ, restrict_apply_univ]
@@ -211,4 +224,214 @@ theorem set_average_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞)
     ENNReal.one_toReal, one_smul]
 #align measure_theory.set_average_const MeasureTheory.set_average_const
 
+-- porting note: was `@[simp]` but `simp` can prove it
+theorem integral_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :
+    ∫ _, ⨍ a, f a ∂μ ∂μ = ∫ x, f x ∂μ := by simp
+#align measure_theory.integral_average MeasureTheory.integral_average
+
+theorem setAverage_setAverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) (s : Set α) :
+    ∫ _ in s, ⨍ a in s, f a ∂μ ∂μ = ∫ x in s, f x ∂μ :=
+  integral_average _ _
+#align measure_theory.set_integral_set_average MeasureTheory.setAverage_setAverage
+
+theorem integral_sub_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :
+    ∫ x, f x - ⨍ a, f a ∂μ ∂μ = 0 := by
+  by_cases hf : Integrable f μ
+  · rw [integral_sub hf (integrable_const _), integral_average, sub_self]
+  refine integral_undef fun h => hf ?_
+  convert h.add (integrable_const (⨍ a, f a ∂μ))
+  exact (sub_add_cancel _ _).symm
+#align measure_theory.integral_sub_average MeasureTheory.integral_sub_average
+
+theorem setAverage_sub_setAverage (hs : μ s ≠ ∞) (f : α → E) :
+    ∫ x in s, f x - ⨍ a in s, f a ∂μ ∂μ = 0 :=
+  haveI : Fact (μ s < ∞) := ⟨lt_top_iff_ne_top.2 hs⟩
+  integral_sub_average _ _
+#align measure_theory.set_integral_sub_set_average MeasureTheory.setAverage_sub_setAverage
+
+theorem integral_average_sub [IsFiniteMeasure μ] (hf : Integrable f μ) :
+    ∫ x, ⨍ a, f a ∂μ - f x ∂μ = 0 := by
+  rw [integral_sub (integrable_const _) hf, integral_average, sub_self]
+#align measure_theory.integral_average_sub MeasureTheory.integral_average_sub
+
+theorem setAverage_setAverage_sub (hs : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+    ∫ x in s, ⨍ a in s, f a ∂μ - f x ∂μ = 0 :=
+  haveI : Fact (μ s < ∞) := ⟨lt_top_iff_ne_top.2 hs⟩
+  integral_average_sub hf
+#align measure_theory.set_integral_set_average_sub MeasureTheory.setAverage_setAverage_sub
+
+end NormedAddCommGroup
+
+theorem ofReal_average {f : α → ℝ} (hf : Integrable f μ) (hf₀ : 0 ≤ᵐ[μ] f) :
+    ENNReal.ofReal (⨍ x, f x ∂μ) = (∫⁻ x, ENNReal.ofReal (f x) ∂μ) / μ univ := by
+  obtain rfl | hμ := eq_or_ne μ 0
+  · simp
+  · rw [average_eq, smul_eq_mul, ← toReal_inv, ofReal_mul toReal_nonneg,
+      ofReal_toReal (inv_ne_top.2 <| measure_univ_ne_zero.2 hμ),
+      ofReal_integral_eq_lintegral_ofReal hf hf₀, ENNReal.div_eq_inv_mul]
+#align measure_theory.of_real_average MeasureTheory.ofReal_average
+
+theorem ofReal_setAverage {f : α → ℝ} (hf : IntegrableOn f s μ) (hf₀ : 0 ≤ᵐ[μ.restrict s] f) :
+    ENNReal.ofReal (⨍ x in s, f x ∂μ) = (∫⁻ x in s, ENNReal.ofReal (f x) ∂μ) / μ s := by
+  simpa using ofReal_average hf hf₀
+#align measure_theory.of_real_set_average MeasureTheory.ofReal_setAverage
+
+theorem average_toReal {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf' : ∀ᵐ x ∂μ, f x ≠ ∞) :
+    ⨍ x, (f x).toReal ∂μ = ((∫⁻ x, f x ∂μ) / μ univ).toReal := by
+  rw [average_eq, smul_eq_mul, toReal_div, ←
+    integral_toReal hf (hf'.mono fun _ => lt_top_iff_ne_top.2), div_eq_inv_mul]
+#align measure_theory.average_to_real MeasureTheory.average_toReal
+
+theorem setAverage_toReal {f : α → ℝ≥0∞} (hf : AEMeasurable f (μ.restrict s))
+    (hf' : ∀ᵐ x ∂μ.restrict s, f x ≠ ∞) :
+    ⨍ x in s, (f x).toReal ∂μ = ((∫⁻ x in s, f x ∂μ) / μ s).toReal := by
+  simpa using average_toReal hf hf'
+#align measure_theory.set_average_to_real MeasureTheory.setAverage_toReal
+
+/-! ### First moment method -/
+
+section FirstMoment
+
+variable {N : Set α} {f : α → ℝ}
+
+/-- **First moment method**. An integrable function is smaller than its mean on a set of positive
+measure. -/
+theorem measure_le_setAverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+    0 < μ ({x ∈ s | f x ≤ ⨍ a in s, f a ∂μ}) := by
+  refine' pos_iff_ne_zero.2 fun H => _
+  replace H : (μ.restrict s) {x | f x ≤ ⨍ a in s, f a ∂μ} = 0
+  · rwa [restrict_apply₀, inter_comm]
+    exact AEStronglyMeasurable.nullMeasurableSet_le hf.1 aestronglyMeasurable_const
+  haveI := Fact.mk hμ₁.lt_top
+  refine' (integral_sub_average (μ.restrict s) f).not_gt _
+  refine' (set_integral_pos_iff_support_of_nonneg_ae _ _).2 _
+  · refine' eq_bot_mono (measure_mono fun x hx => _) H
+    simp only [Pi.zero_apply, sub_nonneg, mem_compl_iff, mem_setOf_eq, not_le] at hx
+    exact hx.le
+  · exact hf.sub (integrableOn_const.2 <| Or.inr <| lt_top_iff_ne_top.2 hμ₁)
+  · rwa [pos_iff_ne_zero, inter_comm, ← diff_compl, ← diff_inter_self_eq_diff, measure_diff_null]
+    refine' eq_bot_mono (measure_mono _) (measure_inter_eq_zero_of_restrict H)
+    exact inter_subset_inter_left _ fun a ha => (sub_eq_zero.1 <| of_not_not ha).le
+#align measure_theory.measure_le_set_average_pos MeasureTheory.measure_le_setAverage_pos
+
+/-- **First moment method**. An integrable function is greater than its mean on a set of positive
+measure. -/
+theorem measure_setAverage_le_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+    0 < μ ({x ∈ s | ⨍ a in s, f a ∂μ ≤ f x}) := by
+  simpa [integral_neg, neg_div] using measure_le_setAverage_pos hμ hμ₁ hf.neg
+#align measure_theory.measure_set_average_le_pos MeasureTheory.measure_setAverage_le_pos
+
+/-- **First moment method**. The minimum of an integrable function is smaller than its mean. -/
+theorem exists_le_setAverage (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+    ∃ x ∈ s, f x ≤ ⨍ a in s, f a ∂μ :=
+  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_le_setAverage_pos hμ hμ₁ hf).ne'
+  ⟨x, hx, h⟩
+#align measure_theory.exists_le_set_average MeasureTheory.exists_le_setAverage
+
+/-- **First moment method**. The maximum of an integrable function is greater than its mean. -/
+theorem exists_setAverage_le (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
+    ∃ x ∈ s, ⨍ a in s, f a ∂μ ≤ f x :=
+  let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_setAverage_le_pos hμ hμ₁ hf).ne'
+  ⟨x, hx, h⟩
+#align measure_theory.exists_set_average_le MeasureTheory.exists_setAverage_le
+
+section FiniteMeasure
+
+variable [IsFiniteMeasure μ]
+
+/-- **First moment method**. An integrable function is smaller than its mean on a set of positive
+measure. -/
+theorem measure_le_average_pos (hμ : μ ≠ 0) (hf : Integrable f μ) :
+    0 < μ {x | f x ≤ ⨍ a, f a ∂μ} := by
+  simpa using measure_le_setAverage_pos (Measure.measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)
+    hf.integrableOn
+#align measure_theory.measure_le_average_pos MeasureTheory.measure_le_average_pos
+
+/-- **First moment method**. An integrable function is greater than its mean on a set of positive
+measure. -/
+theorem measure_average_le_pos (hμ : μ ≠ 0) (hf : Integrable f μ) :
+    0 < μ {x | ⨍ a, f a ∂μ ≤ f x} := by
+  simpa using measure_setAverage_le_pos (Measure.measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)
+    hf.integrableOn
+#align measure_theory.measure_average_le_pos MeasureTheory.measure_average_le_pos
+
+/-- **First moment method**. The minimum of an integrable function is smaller than its mean. -/
+theorem exists_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) : ∃ x, f x ≤ ⨍ a, f a ∂μ :=
+  let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_le_average_pos hμ hf).ne'
+  ⟨x, hx⟩
+#align measure_theory.exists_le_average MeasureTheory.exists_le_average
+
+/-- **First moment method**. The maximum of an integrable function is greater than its mean. -/
+theorem exists_average_le (hμ : μ ≠ 0) (hf : Integrable f μ) : ∃ x, ⨍ a, f a ∂μ ≤ f x :=
+  let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_average_le_pos hμ hf).ne'
+  ⟨x, hx⟩
+#align measure_theory.exists_average_le MeasureTheory.exists_average_le
+
+/-- **First moment method**. The minimum of an integrable function is smaller than its mean, while
+avoiding a null set. -/
+theorem exists_not_mem_null_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) (hN : μ N = 0) :
+    ∃ x, x ∉ N ∧ f x ≤ ⨍ a, f a ∂μ := by
+  have := measure_le_average_pos hμ hf
+  rw [← measure_diff_null hN] at this
+  obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
+  exact ⟨x, hxN, hx⟩
+#align measure_theory.exists_not_mem_null_le_average MeasureTheory.exists_not_mem_null_le_average
+
+/-- **First moment method**. The maximum of an integrable function is greater than its mean, while
+avoiding a null set. -/
+theorem exists_not_mem_null_average_le (hμ : μ ≠ 0) (hf : Integrable f μ) (hN : μ N = 0) :
+    ∃ x, x ∉ N ∧ ⨍ a, f a ∂μ ≤ f x := by
+  simpa [integral_neg, neg_div] using exists_not_mem_null_le_average hμ hf.neg hN
+#align measure_theory.exists_not_mem_null_average_le MeasureTheory.exists_not_mem_null_average_le
+
+end FiniteMeasure
+
+section ProbabilityMeasure
+
+variable [IsProbabilityMeasure μ]
+
+/-- **First moment method**. An integrable function is smaller than its integral on a set of
+positive measure. -/
+theorem measure_le_integral_pos (hf : Integrable f μ) : 0 < μ {x | f x ≤ ∫ a, f a ∂μ} := by
+  simpa only [average_eq_integral] using
+    measure_le_average_pos (IsProbabilityMeasure.ne_zero μ) hf
+#align measure_theory.measure_le_integral_pos MeasureTheory.measure_le_integral_pos
+
+/-- **First moment method**. An integrable function is greater than its integral on a set of
+positive measure. -/
+theorem measure_integral_le_pos (hf : Integrable f μ) : 0 < μ {x | ∫ a, f a ∂μ ≤ f x} := by
+  simpa only [average_eq_integral] using
+    measure_average_le_pos (IsProbabilityMeasure.ne_zero μ) hf
+#align measure_theory.measure_integral_le_pos MeasureTheory.measure_integral_le_pos
+
+/-- **First moment method**. The minimum of an integrable function is smaller than its integral. -/
+theorem exists_le_integral (hf : Integrable f μ) : ∃ x, f x ≤ ∫ a, f a ∂μ := by
+  simpa only [average_eq_integral] using exists_le_average (IsProbabilityMeasure.ne_zero μ) hf
+#align measure_theory.exists_le_integral MeasureTheory.exists_le_integral
+
+/-- **First moment method**. The maximum of an integrable function is greater than its integral. -/
+theorem exists_integral_le (hf : Integrable f μ) : ∃ x, ∫ a, f a ∂μ ≤ f x := by
+  simpa only [average_eq_integral] using exists_average_le (IsProbabilityMeasure.ne_zero μ) hf
+#align measure_theory.exists_integral_le MeasureTheory.exists_integral_le
+
+/-- **First moment method**. The minimum of an integrable function is smaller than its integral,
+while avoiding a null set. -/
+theorem exists_not_mem_null_le_integral (hf : Integrable f μ) (hN : μ N = 0) :
+    ∃ x, x ∉ N ∧ f x ≤ ∫ a, f a ∂μ := by
+  simpa only [average_eq_integral] using
+    exists_not_mem_null_le_average (IsProbabilityMeasure.ne_zero μ) hf hN
+#align measure_theory.exists_not_mem_null_le_integral MeasureTheory.exists_not_mem_null_le_integral
+
+/-- **First moment method**. The maximum of an integrable function is greater than its integral,
+while avoiding a null set. -/
+theorem exists_not_mem_null_integral_le (hf : Integrable f μ) (hN : μ N = 0) :
+    ∃ x, x ∉ N ∧ ∫ a, f a ∂μ ≤ f x := by
+  simpa only [average_eq_integral] using
+    exists_not_mem_null_average_le (IsProbabilityMeasure.ne_zero μ) hf hN
+#align measure_theory.exists_not_mem_null_integral_le MeasureTheory.exists_not_mem_null_integral_le
+
+end ProbabilityMeasure
+
+end FirstMoment
+
 end MeasureTheory

fix: change compl precedence (#5586)

Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

4d4f0f25

Diff

@@ -189,7 +189,7 @@ theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint
 #align measure_theory.average_union_mem_segment MeasureTheory.average_union_mem_segment
 
 theorem average_mem_openSegment_compl_self [IsFiniteMeasure μ] {f : α → E} {s : Set α}
-    (hs : NullMeasurableSet s μ) (hs₀ : μ s ≠ 0) (hsc₀ : μ (sᶜ) ≠ 0) (hfi : Integrable f μ) :
+    (hs : NullMeasurableSet s μ) (hs₀ : μ s ≠ 0) (hsc₀ : μ sᶜ ≠ 0) (hfi : Integrable f μ) :
     ⨍ x, f x ∂μ ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in sᶜ, f x ∂μ) := by
   simpa only [union_compl_self, restrict_univ] using
     average_union_mem_openSegment aedisjoint_compl_right hs.compl hs₀ hsc₀ (measure_ne_top _ _)

chore: copy integral notation precedence from mathport (#4963)

Also drop some of the parentheses that are no longer needed.

60fe1293

Diff

@@ -67,16 +67,20 @@ noncomputable def average (f : α → E) :=
   ∫ x, f x ∂(μ univ)⁻¹ • μ
 #align measure_theory.average MeasureTheory.average
 
-notation3"⨍ "(...)", "r:(scoped f => f)" ∂"μ => average μ r
+@[inherit_doc average]
+notation3 "⨍ "(...)", "r:60:(scoped f => f)" ∂"μ:70 => average μ r
 
-notation3"⨍ "(...)", "r:(scoped f => average volume f) => r
+@[inherit_doc average]
+notation3 "⨍ "(...)", "r:60:(scoped f => average volume f) => r
 
-notation3"⨍ "(...)" in "s", "r:(scoped f => f)" ∂"μ => average (Measure.restrict μ s) r
+@[inherit_doc average]
+notation3 "⨍ "(...)" in "s", "r:60:(scoped f => f)" ∂"μ:70 => average (Measure.restrict μ s) r
 
-notation3"⨍ "(...)" in "s", "r:(scoped f => average (Measure.restrict volume s) f) => r
+@[inherit_doc average]
+notation3 "⨍ "(...)" in "s", "r:60:(scoped f => average (Measure.restrict volume s) f) => r
 
 @[simp]
-theorem average_zero : (⨍ _, (0 : E) ∂μ) = 0 := by rw [average, integral_zero]
+theorem average_zero : ⨍ _, (0 : E) ∂μ = 0 := by rw [average, integral_zero]
 #align measure_theory.average_zero MeasureTheory.average_zero
 
 @[simp]
@@ -85,19 +89,19 @@ theorem average_zero_measure (f : α → E) : (⨍ x, f x ∂(0 : Measure α)) =
 #align measure_theory.average_zero_measure MeasureTheory.average_zero_measure
 
 @[simp]
-theorem average_neg (f : α → E) : (⨍ x, -f x ∂μ) = -⨍ x, f x ∂μ :=
+theorem average_neg (f : α → E) : ⨍ x, -f x ∂μ = -⨍ x, f x ∂μ :=
   integral_neg f
 #align measure_theory.average_neg MeasureTheory.average_neg
 
-theorem average_eq' (f : α → E) : (⨍ x, f x ∂μ) = ∫ x, f x ∂(μ univ)⁻¹ • μ :=
+theorem average_eq' (f : α → E) : ⨍ x, f x ∂μ = ∫ x, f x ∂(μ univ)⁻¹ • μ :=
   rfl
 #align measure_theory.average_eq' MeasureTheory.average_eq'
 
-theorem average_eq (f : α → E) : (⨍ x, f x ∂μ) = (μ univ).toReal⁻¹ • ∫ x, f x ∂μ := by
+theorem average_eq (f : α → E) : ⨍ x, f x ∂μ = (μ univ).toReal⁻¹ • ∫ x, f x ∂μ := by
   rw [average_eq', integral_smul_measure, ENNReal.toReal_inv]
 #align measure_theory.average_eq MeasureTheory.average_eq
 
-theorem average_eq_integral [IsProbabilityMeasure μ] (f : α → E) : (⨍ x, f x ∂μ) = ∫ x, f x ∂μ := by
+theorem average_eq_integral [IsProbabilityMeasure μ] (f : α → E) : ⨍ x, f x ∂μ = ∫ x, f x ∂μ := by
   rw [average, measure_univ, inv_one, one_smul]
 #align measure_theory.average_eq_integral MeasureTheory.average_eq_integral
 
@@ -112,24 +116,24 @@ theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
 #align measure_theory.measure_smul_average MeasureTheory.measure_smul_average
 
 theorem set_average_eq (f : α → E) (s : Set α) :
-    (⨍ x in s, f x ∂μ) = (μ s).toReal⁻¹ • ∫ x in s, f x ∂μ := by
+    ⨍ x in s, f x ∂μ = (μ s).toReal⁻¹ • ∫ x in s, f x ∂μ := by
   rw [average_eq, restrict_apply_univ]
 #align measure_theory.set_average_eq MeasureTheory.set_average_eq
 
 theorem set_average_eq' (f : α → E) (s : Set α) :
-    (⨍ x in s, f x ∂μ) = ∫ x, f x ∂(μ s)⁻¹ • μ.restrict s := by
+    ⨍ x in s, f x ∂μ = ∫ x, f x ∂(μ s)⁻¹ • μ.restrict s := by
   simp only [average_eq', restrict_apply_univ]
 #align measure_theory.set_average_eq' MeasureTheory.set_average_eq'
 
 variable {μ}
 
-theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : (⨍ x, f x ∂μ) = ⨍ x, g x ∂μ := by
+theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : ⨍ x, f x ∂μ = ⨍ x, g x ∂μ := by
   simp only [average_eq, integral_congr_ae h]
 #align measure_theory.average_congr MeasureTheory.average_congr
 
 theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E}
     (hμ : Integrable f μ) (hν : Integrable f ν) :
-    (⨍ x, f x ∂μ + ν) =
+    ⨍ x, f x ∂(μ + ν) =
       (((μ univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂μ) +
         ((ν univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂ν := by
   simp only [div_eq_inv_mul, mul_smul, measure_smul_average, ← smul_add, ←
@@ -138,7 +142,7 @@ theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeas
 #align measure_theory.average_add_measure MeasureTheory.average_add_measure
 
 theorem average_pair {f : α → E} {g : α → F} (hfi : Integrable f μ) (hgi : Integrable g μ) :
-    (⨍ x, (f x, g x) ∂μ) = (⨍ x, f x ∂μ, ⨍ x, g x ∂μ) :=
+    ⨍ x, (f x, g x) ∂μ = (⨍ x, f x ∂μ, ⨍ x, g x ∂μ) :=
   integral_pair hfi.to_average hgi.to_average
 #align measure_theory.average_pair MeasureTheory.average_pair
 
@@ -150,7 +154,7 @@ theorem measure_smul_set_average (f : α → E) {s : Set α} (h : μ s ≠ ∞)
 
 theorem average_union {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
     (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
-    (⨍ x in s ∪ t, f x ∂μ) =
+    ⨍ x in s ∪ t, f x ∂μ =
       (((μ s).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ x in s, f x ∂μ) +
         ((μ t).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ x in t, f x ∂μ := by
   haveI := Fact.mk hsμ.lt_top; haveI := Fact.mk htμ.lt_top
@@ -160,7 +164,7 @@ theorem average_union {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t) (ht
 theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t)
     (ht : NullMeasurableSet t μ) (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞)
     (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
-    (⨍ x in s ∪ t, f x ∂μ) ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in t, f x ∂μ) := by
+    ⨍ x in s ∪ t, f x ∂μ ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in t, f x ∂μ) := by
   replace hs₀ : 0 < (μ s).toReal; exact ENNReal.toReal_pos hs₀ hsμ
   replace ht₀ : 0 < (μ t).toReal; exact ENNReal.toReal_pos ht₀ htμ
   refine' mem_openSegment_iff_div.mpr
@@ -170,7 +174,7 @@ theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AEDisj
 theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t)
     (ht : NullMeasurableSet t μ) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ)
     (hft : IntegrableOn f t μ) :
-    (⨍ x in s ∪ t, f x ∂μ) ∈ [⨍ x in s, f x ∂μ -[ℝ] ⨍ x in t, f x ∂μ] := by
+    ⨍ x in s ∪ t, f x ∂μ ∈ [⨍ x in s, f x ∂μ -[ℝ] ⨍ x in t, f x ∂μ] := by
   by_cases hse : μ s = 0
   · rw [← ae_eq_empty] at hse
     rw [restrict_congr_set (hse.union EventuallyEq.rfl), empty_union]
@@ -186,14 +190,14 @@ theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint
 
 theorem average_mem_openSegment_compl_self [IsFiniteMeasure μ] {f : α → E} {s : Set α}
     (hs : NullMeasurableSet s μ) (hs₀ : μ s ≠ 0) (hsc₀ : μ (sᶜ) ≠ 0) (hfi : Integrable f μ) :
-    (⨍ x, f x ∂μ) ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in sᶜ, f x ∂μ) := by
+    ⨍ x, f x ∂μ ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in sᶜ, f x ∂μ) := by
   simpa only [union_compl_self, restrict_univ] using
     average_union_mem_openSegment aedisjoint_compl_right hs.compl hs₀ hsc₀ (measure_ne_top _ _)
       (measure_ne_top _ _) hfi.integrableOn hfi.integrableOn
 #align measure_theory.average_mem_open_segment_compl_self MeasureTheory.average_mem_openSegment_compl_self
 
 @[simp]
-theorem average_const [IsFiniteMeasure μ] [h : μ.ae.NeBot] (c : E) : (⨍ _, c ∂μ) = c := by
+theorem average_const [IsFiniteMeasure μ] [h : μ.ae.NeBot] (c : E) : ⨍ _, c ∂μ = c := by
   simp only [average_eq, integral_const, Measure.restrict_apply, MeasurableSet.univ, one_smul,
     univ_inter, smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel,
     measure_ne_top μ univ, Ne.def, measure_univ_eq_zero, ae_neBot.1 h, not_false_iff,
@@ -201,7 +205,7 @@ theorem average_const [IsFiniteMeasure μ] [h : μ.ae.NeBot] (c : E) : (⨍ _, c
 #align measure_theory.average_const MeasureTheory.average_const
 
 theorem set_average_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : E) :
-    (⨍ _ in s, c ∂μ) = c := by
+    ⨍ _ in s, c ∂μ = c := by
   simp only [set_average_eq, integral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter,
     smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel hs₀ hs,
     ENNReal.one_toReal, one_smul]