analysis.convex.integral ⟷ Mathlib.Analysis.Convex.Integral

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

@@ -408,17 +408,17 @@ theorem ae_eq_const_or_norm_integral_lt_of_norm_le_const [StrictConvexSpace ℝ
 #align ae_eq_const_or_norm_integral_lt_of_norm_le_const ae_eq_const_or_norm_integral_lt_of_norm_le_const
 -/
 
-#print ae_eq_const_or_norm_set_integral_lt_of_norm_le_const /-
+#print ae_eq_const_or_norm_setIntegral_lt_of_norm_le_const /-
 /-- If `E` is a strictly convex normed space and `f : α → E` is a function such that `‖f x‖ ≤ C`
 a.e. on a set `t` of finite measure, then either this function is a.e. equal to its average value on
 `t`, or the norm of its integral over `t` is strictly less than `(μ t).to_real * C`. -/
-theorem ae_eq_const_or_norm_set_integral_lt_of_norm_le_const [StrictConvexSpace ℝ E] (ht : μ t ≠ ∞)
+theorem ae_eq_const_or_norm_setIntegral_lt_of_norm_le_const [StrictConvexSpace ℝ E] (ht : μ t ≠ ∞)
     (h_le : ∀ᵐ x ∂μ.restrict t, ‖f x‖ ≤ C) :
     f =ᵐ[μ.restrict t] const α (⨍ x in t, f x ∂μ) ∨ ‖∫ x in t, f x ∂μ‖ < (μ t).toReal * C :=
   by
   haveI := Fact.mk ht.lt_top
   rw [← restrict_apply_univ]
   exact ae_eq_const_or_norm_integral_lt_of_norm_le_const h_le
-#align ae_eq_const_or_norm_set_integral_lt_of_norm_le_const ae_eq_const_or_norm_set_integral_lt_of_norm_le_const
+#align ae_eq_const_or_norm_set_integral_lt_of_norm_le_const ae_eq_const_or_norm_setIntegral_lt_of_norm_le_const
 -/
 

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

@@ -5,7 +5,7 @@ Authors: Yury G. Kudryashov
 -/
 import Analysis.Convex.Function
 import Analysis.Convex.StrictConvexSpace
-import MeasureTheory.Function.AeEqOfIntegral
+import MeasureTheory.Function.AEEqOfIntegral
 import MeasureTheory.Integral.Average
 
 #align_import analysis.convex.integral from "leanprover-community/mathlib"@"c20927220ef87bb4962ba08bf6da2ce3cf50a6dd"

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

@@ -68,7 +68,7 @@ theorem Convex.integral_mem [IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc :
     by
     rcases(hf.and hfg).exists with ⟨x₀, h₀⟩
     exact ⟨f x₀, by simp only [h₀.2, mem_range_self], h₀.1⟩
-  rw [integral_congr_ae hfg]; rw [integrable_congr hfg] at hfi 
+  rw [integral_congr_ae hfg]; rw [integrable_congr hfg] at hfi
   have hg : ∀ᵐ x ∂μ, g x ∈ closure (range g ∩ s) :=
     by
     filter_upwards [hfg.rw (fun x y => y ∈ s) hf] with x hx
@@ -274,16 +274,16 @@ theorem ae_eq_const_or_exists_average_ne_compl [IsFiniteMeasure μ] (hfi : Integ
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨
       ∃ t, MeasurableSet t ∧ μ t ≠ 0 ∧ μ (tᶜ) ≠ 0 ∧ ⨍ x in t, f x ∂μ ≠ ⨍ x in tᶜ, f x ∂μ :=
   by
-  refine' or_iff_not_imp_right.mpr fun H => _; push_neg at H 
+  refine' or_iff_not_imp_right.mpr fun H => _; push_neg at H
   refine' hfi.ae_eq_of_forall_set_integral_eq _ _ (integrable_const _) fun t ht ht' => _; clear ht'
   simp only [const_apply, set_integral_const]
   by_cases h₀ : μ t = 0
   · rw [restrict_eq_zero.2 h₀, integral_zero_measure, h₀, ENNReal.zero_toReal, zero_smul]
   by_cases h₀' : μ (tᶜ) = 0
-  · rw [← ae_eq_univ] at h₀' 
+  · rw [← ae_eq_univ] at h₀'
     rw [restrict_congr_set h₀', restrict_univ, measure_congr h₀', measure_smul_average]
   have := average_mem_open_segment_compl_self ht.null_measurable_set h₀ h₀' hfi
-  rw [← H t ht h₀ h₀', openSegment_same, mem_singleton_iff] at this 
+  rw [← H t ht h₀ h₀', openSegment_same, mem_singleton_iff] at this
   rw [this, measure_smul_set_average _ (measure_ne_top μ _)]
 #align ae_eq_const_or_exists_average_ne_compl ae_eq_const_or_exists_average_ne_compl
 -/
@@ -294,12 +294,11 @@ positive measure, the average value of `f` over `t` belongs to the interior of `
 of `f` over the whole space belongs to the interior of `s`. -/
 theorem Convex.average_mem_interior_of_set [IsFiniteMeasure μ] (hs : Convex ℝ s) (h0 : μ t ≠ 0)
     (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) (ht : ⨍ x in t, f x ∂μ ∈ interior s) :
-    ⨍ x, f x ∂μ ∈ interior s :=
-  by
-  rw [← measure_to_measurable] at h0 ; rw [← restrict_to_measurable (measure_ne_top μ t)] at ht 
+    ⨍ x, f x ∂μ ∈ interior s := by
+  rw [← measure_to_measurable] at h0; rw [← restrict_to_measurable (measure_ne_top μ t)] at ht
   by_cases h0' : μ (to_measurable μ tᶜ) = 0
-  · rw [← ae_eq_univ] at h0' 
-    rwa [restrict_congr_set h0', restrict_univ] at ht 
+  · rw [← ae_eq_univ] at h0'
+    rwa [restrict_congr_set h0', restrict_univ] at ht
   exact
     hs.open_segment_interior_closure_subset_interior ht
       (hs.set_average_mem_closure h0' (measure_ne_top _ _) (ae_restrict_of_ae hfs)
@@ -345,7 +344,7 @@ theorem StrictConvexOn.ae_eq_const_or_map_average_lt [IsFiniteMeasure μ] (hg :
   rcases average_mem_open_segment_compl_self hm.null_measurable_set h₀ h₀' (hfi.prod_mk hgi) with
     ⟨a, b, ha, hb, hab, h_avg⟩
   simp only [average_pair hfi hgi, average_pair hfi.integrable_on hgi.integrable_on, Prod.smul_mk,
-    Prod.mk_add_mk, Prod.mk.inj_iff, (· ∘ ·)] at h_avg 
+    Prod.mk_add_mk, Prod.mk.inj_iff, (· ∘ ·)] at h_avg
   rw [← h_avg.1, ← h_avg.2]
   calc
     g (a • ⨍ x in t, f x ∂μ + b • ⨍ x in tᶜ, f x ∂μ) <
@@ -405,7 +404,7 @@ theorem ae_eq_const_or_norm_integral_lt_of_norm_le_const [StrictConvexSpace ℝ
     simp [ENNReal.toReal_pos_iff, pos_iff_ne_zero, h₀, measure_lt_top]
   refine' (ae_eq_const_or_norm_average_lt_of_norm_le_const h_le).imp_right fun H => _
   rwa [average_eq, norm_smul, norm_inv, Real.norm_eq_abs, abs_of_pos hμ, ← div_eq_inv_mul,
-    div_lt_iff' hμ] at H 
+    div_lt_iff' hμ] at H
 #align ae_eq_const_or_norm_integral_lt_of_norm_le_const ae_eq_const_or_norm_integral_lt_of_norm_le_const
 -/
 

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

@@ -3,10 +3,10 @@ Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov
 -/
-import Mathbin.Analysis.Convex.Function
-import Mathbin.Analysis.Convex.StrictConvexSpace
-import Mathbin.MeasureTheory.Function.AeEqOfIntegral
-import Mathbin.MeasureTheory.Integral.Average
+import Analysis.Convex.Function
+import Analysis.Convex.StrictConvexSpace
+import MeasureTheory.Function.AeEqOfIntegral
+import MeasureTheory.Integral.Average
 
 #align_import analysis.convex.integral from "leanprover-community/mathlib"@"c20927220ef87bb4962ba08bf6da2ce3cf50a6dd"
 

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

@@ -2,17 +2,14 @@
 Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
 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 analysis.convex.integral
-! leanprover-community/mathlib commit c20927220ef87bb4962ba08bf6da2ce3cf50a6dd
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Convex.Function
 import Mathbin.Analysis.Convex.StrictConvexSpace
 import Mathbin.MeasureTheory.Function.AeEqOfIntegral
 import Mathbin.MeasureTheory.Integral.Average
 
+#align_import analysis.convex.integral from "leanprover-community/mathlib"@"c20927220ef87bb4962ba08bf6da2ce3cf50a6dd"
+
 /-!
 # Jensen's inequality for integrals
 

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

@@ -56,6 +56,7 @@ variable {α E F : Type _} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [Nor
 -/
 
 
+#print Convex.integral_mem /-
 /-- If `μ` is a probability measure on `α`, `s` is a convex closed set in `E`, and `f` is an
 integrable function sending `μ`-a.e. points to `s`, then the expected value of `f` belongs to `s`:
 `∫ x, f x ∂μ ∈ s`. See also `convex.sum_mem` for a finite sum version of this lemma. -/
@@ -89,7 +90,9 @@ theorem Convex.integral_mem [IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc :
     apply inter_subset_right (range g)
     exact simple_func.approx_on_mem hgm.measurable _ _ _
 #align convex.integral_mem Convex.integral_mem
+-/
 
+#print Convex.average_mem /-
 /-- If `μ` is a non-zero finite measure on `α`, `s` is a convex closed set in `E`, and `f` is an
 integrable function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `s`:
 `⨍ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem` for a finite sum version of this lemma. -/
@@ -100,7 +103,9 @@ theorem Convex.average_mem [IsFiniteMeasure μ] (hs : Convex ℝ s) (hsc : IsClo
   refine' hs.integral_mem hsc (ae_mono' _ hfs) hfi.to_average
   exact absolutely_continuous.smul (refl _) _
 #align convex.average_mem Convex.average_mem
+-/
 
+#print Convex.set_average_mem /-
 /-- If `μ` is a non-zero finite measure on `α`, `s` is a convex closed set in `E`, and `f` is an
 integrable function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `s`:
 `⨍ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem` for a finite sum version of this lemma. -/
@@ -111,7 +116,9 @@ theorem Convex.set_average_mem (hs : Convex ℝ s) (hsc : IsClosed s) (h0 : μ t
   refine' hs.average_mem hsc _ hfs hfi
   rwa [Ne.def, restrict_eq_zero]
 #align convex.set_average_mem Convex.set_average_mem
+-/
 
+#print Convex.set_average_mem_closure /-
 /-- If `μ` is a non-zero finite measure on `α`, `s` is a convex set in `E`, and `f` is an integrable
 function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `closure s`:
 `⨍ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem` for a finite sum version of this lemma. -/
@@ -119,7 +126,9 @@ theorem Convex.set_average_mem_closure (hs : Convex ℝ s) (h0 : μ t ≠ 0) (ht
     (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s) (hfi : IntegrableOn f t μ) : ⨍ x in t, f x ∂μ ∈ closure s :=
   hs.closure.set_average_mem isClosed_closure h0 ht (hfs.mono fun x hx => subset_closure hx) hfi
 #align convex.set_average_mem_closure Convex.set_average_mem_closure
+-/
 
+#print ConvexOn.average_mem_epigraph /-
 theorem ConvexOn.average_mem_epigraph [IsFiniteMeasure μ] (hg : ConvexOn ℝ s g)
     (hgc : ContinuousOn g s) (hsc : IsClosed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
     (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
@@ -130,7 +139,9 @@ theorem ConvexOn.average_mem_epigraph [IsFiniteMeasure μ] (hg : ConvexOn ℝ s
   simpa only [average_pair hfi hgi] using
     hg.convex_epigraph.average_mem (hsc.epigraph hgc) hμ ht_mem (hfi.prod_mk hgi)
 #align convex_on.average_mem_epigraph ConvexOn.average_mem_epigraph
+-/
 
+#print ConcaveOn.average_mem_hypograph /-
 theorem ConcaveOn.average_mem_hypograph [IsFiniteMeasure μ] (hg : ConcaveOn ℝ s g)
     (hgc : ContinuousOn g s) (hsc : IsClosed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
     (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
@@ -138,7 +149,9 @@ theorem ConcaveOn.average_mem_hypograph [IsFiniteMeasure μ] (hg : ConcaveOn ℝ
   simpa only [mem_set_of_eq, Pi.neg_apply, average_neg, neg_le_neg_iff] using
     hg.neg.average_mem_epigraph hgc.neg hsc hμ hfs hfi hgi.neg
 #align concave_on.average_mem_hypograph ConcaveOn.average_mem_hypograph
+-/
 
+#print ConvexOn.map_average_le /-
 /-- **Jensen's inequality**: if a function `g : E → ℝ` is convex and continuous on a convex closed
 set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a function sending
 `μ`-a.e. points to `s`, then the value of `g` at the average value of `f` is less than or equal to
@@ -149,7 +162,9 @@ theorem ConvexOn.map_average_le [IsFiniteMeasure μ] (hg : ConvexOn ℝ s g) (hg
     (hgi : Integrable (g ∘ f) μ) : g (⨍ x, f x ∂μ) ≤ ⨍ x, g (f x) ∂μ :=
   (hg.average_mem_epigraph hgc hsc hμ hfs hfi hgi).2
 #align convex_on.map_average_le ConvexOn.map_average_le
+-/
 
+#print ConcaveOn.le_map_average /-
 /-- **Jensen's inequality**: if a function `g : E → ℝ` is concave and continuous on a convex closed
 set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a function sending
 `μ`-a.e. points to `s`, then the average value of `g ∘ f` is less than or equal to the value of `g`
@@ -160,7 +175,9 @@ theorem ConcaveOn.le_map_average [IsFiniteMeasure μ] (hg : ConcaveOn ℝ s g) (
     (hgi : Integrable (g ∘ f) μ) : ⨍ x, g (f x) ∂μ ≤ g (⨍ x, f x ∂μ) :=
   (hg.average_mem_hypograph hgc hsc hμ hfs hfi hgi).2
 #align concave_on.le_map_average ConcaveOn.le_map_average
+-/
 
+#print ConvexOn.set_average_mem_epigraph /-
 /-- **Jensen's inequality**: if a function `g : E → ℝ` is convex and continuous on a convex closed
 set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a function sending
 `μ`-a.e. points of a set `t` to `s`, then the value of `g` at the average value of `f` over `t` is
@@ -175,7 +192,9 @@ theorem ConvexOn.set_average_mem_epigraph (hg : ConvexOn ℝ s g) (hgc : Continu
   refine' hg.average_mem_epigraph hgc hsc _ hfs hfi hgi
   rwa [Ne.def, restrict_eq_zero]
 #align convex_on.set_average_mem_epigraph ConvexOn.set_average_mem_epigraph
+-/
 
+#print ConcaveOn.set_average_mem_hypograph /-
 /-- **Jensen's inequality**: if a function `g : E → ℝ` is concave and continuous on a convex closed
 set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a function sending
 `μ`-a.e. points of a set `t` to `s`, then the average value of `g ∘ f` over `t` is less than or
@@ -188,7 +207,9 @@ theorem ConcaveOn.set_average_mem_hypograph (hg : ConcaveOn ℝ s g) (hgc : Cont
   simpa only [mem_set_of_eq, Pi.neg_apply, average_neg, neg_le_neg_iff] using
     hg.neg.set_average_mem_epigraph hgc.neg hsc h0 ht hfs hfi hgi.neg
 #align concave_on.set_average_mem_hypograph ConcaveOn.set_average_mem_hypograph
+-/
 
+#print ConvexOn.map_set_average_le /-
 /-- **Jensen's inequality**: if a function `g : E → ℝ` is convex and continuous on a convex closed
 set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a function sending
 `μ`-a.e. points of a set `t` to `s`, then the value of `g` at the average value of `f` over `t` is
@@ -200,7 +221,9 @@ theorem ConvexOn.map_set_average_le (hg : ConvexOn ℝ s g) (hgc : ContinuousOn
     g (⨍ x in t, f x ∂μ) ≤ ⨍ x in t, g (f x) ∂μ :=
   (hg.set_average_mem_epigraph hgc hsc h0 ht hfs hfi hgi).2
 #align convex_on.map_set_average_le ConvexOn.map_set_average_le
+-/
 
+#print ConcaveOn.le_map_set_average /-
 /-- **Jensen's inequality**: if a function `g : E → ℝ` is concave and continuous on a convex closed
 set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a function sending
 `μ`-a.e. points of a set `t` to `s`, then the average value of `g ∘ f` over `t` is less than or
@@ -212,7 +235,9 @@ theorem ConcaveOn.le_map_set_average (hg : ConcaveOn ℝ s g) (hgc : ContinuousO
     ⨍ x in t, g (f x) ∂μ ≤ g (⨍ x in t, f x ∂μ) :=
   (hg.set_average_mem_hypograph hgc hsc h0 ht hfs hfi hgi).2
 #align concave_on.le_map_set_average ConcaveOn.le_map_set_average
+-/
 
+#print ConvexOn.map_integral_le /-
 /-- **Jensen's inequality**: if a function `g : E → ℝ` is convex and continuous on a convex closed
 set `s`, `μ` is a probability measure on `α`, and `f : α → E` is a function sending `μ`-a.e.  points
 to `s`, then the value of `g` at the expected value of `f` is less than or equal to the expected
@@ -224,7 +249,9 @@ theorem ConvexOn.map_integral_le [IsProbabilityMeasure μ] (hg : ConvexOn ℝ s
   simpa only [average_eq_integral] using
     hg.map_average_le hgc hsc (is_probability_measure.ne_zero μ) hfs hfi hgi
 #align convex_on.map_integral_le ConvexOn.map_integral_le
+-/
 
+#print ConcaveOn.le_map_integral /-
 /-- **Jensen's inequality**: if a function `g : E → ℝ` is concave and continuous on a convex closed
 set `s`, `μ` is a probability measure on `α`, and `f : α → E` is a function sending `μ`-a.e.  points
 to `s`, then the expected value of `g ∘ f` is less than or equal to the value of `g` at the expected
@@ -235,12 +262,14 @@ theorem ConcaveOn.le_map_integral [IsProbabilityMeasure μ] (hg : ConcaveOn ℝ
   simpa only [average_eq_integral] using
     hg.le_map_average hgc hsc (is_probability_measure.ne_zero μ) hfs hfi hgi
 #align concave_on.le_map_integral ConcaveOn.le_map_integral
+-/
 
 /-!
 ### Strict Jensen's inequality
 -/
 
 
+#print ae_eq_const_or_exists_average_ne_compl /-
 /-- If `f : α → E` is an integrable function, then either it is a.e. equal to the constant
 `⨍ x, f x ∂μ` or there exists a measurable set such that `μ t ≠ 0`, `μ tᶜ ≠ 0`, and the average
 values of `f` over `t` and `tᶜ` are different. -/
@@ -260,7 +289,9 @@ theorem ae_eq_const_or_exists_average_ne_compl [IsFiniteMeasure μ] (hfi : Integ
   rw [← H t ht h₀ h₀', openSegment_same, mem_singleton_iff] at this 
   rw [this, measure_smul_set_average _ (measure_ne_top μ _)]
 #align ae_eq_const_or_exists_average_ne_compl ae_eq_const_or_exists_average_ne_compl
+-/
 
+#print Convex.average_mem_interior_of_set /-
 /-- If an integrable function `f : α → E` takes values in a convex set `s` and for some set `t` of
 positive measure, the average value of `f` over `t` belongs to the interior of `s`, then the average
 of `f` over the whole space belongs to the interior of `s`. -/
@@ -279,7 +310,9 @@ theorem Convex.average_mem_interior_of_set [IsFiniteMeasure μ] (hs : Convex ℝ
       (average_mem_open_segment_compl_self (measurable_set_to_measurable μ t).NullMeasurableSet h0
         h0' hfi)
 #align convex.average_mem_interior_of_set Convex.average_mem_interior_of_set
+-/
 
+#print StrictConvex.ae_eq_const_or_average_mem_interior /-
 /-- If an integrable function `f : α → E` takes values in a strictly convex closed set `s`, then
 either it is a.e. equal to its average value, or its average value belongs to the interior of
 `s`. -/
@@ -295,7 +328,9 @@ theorem StrictConvex.ae_eq_const_or_average_mem_interior [IsFiniteMeasure μ] (h
     hs.open_segment_subset (this h₀) (this h₀') hne
       (average_mem_open_segment_compl_self hm.null_measurable_set h₀ h₀' hfi)
 #align strict_convex.ae_eq_const_or_average_mem_interior StrictConvex.ae_eq_const_or_average_mem_interior
+-/
 
+#print StrictConvexOn.ae_eq_const_or_map_average_lt /-
 /-- **Jensen's inequality**, strict version: if an integrable function `f : α → E` takes values in a
 convex closed set `s`, and `g : E → ℝ` is continuous and strictly convex on `s`, then
 either `f` is a.e. equal to its average value, or `g (⨍ x, f x ∂μ) < ⨍ x, g (f x) ∂μ`. -/
@@ -323,7 +358,9 @@ theorem StrictConvexOn.ae_eq_const_or_map_average_lt [IsFiniteMeasure μ] (hg :
       add_le_add (mul_le_mul_of_nonneg_left (this h₀).2 ha.le)
         (mul_le_mul_of_nonneg_left (this h₀').2 hb.le)
 #align strict_convex_on.ae_eq_const_or_map_average_lt StrictConvexOn.ae_eq_const_or_map_average_lt
+-/
 
+#print StrictConcaveOn.ae_eq_const_or_lt_map_average /-
 /-- **Jensen's inequality**, strict version: if an integrable function `f : α → E` takes values in a
 convex closed set `s`, and `g : E → ℝ` is continuous and strictly concave on `s`, then
 either `f` is a.e. equal to its average value, or `⨍ x, g (f x) ∂μ < g (⨍ x, f x ∂μ)`. -/
@@ -334,7 +371,9 @@ theorem StrictConcaveOn.ae_eq_const_or_lt_map_average [IsFiniteMeasure μ]
   simpa only [Pi.neg_apply, average_neg, neg_lt_neg_iff] using
     hg.neg.ae_eq_const_or_map_average_lt hgc.neg hsc hfs hfi hgi.neg
 #align strict_concave_on.ae_eq_const_or_lt_map_average StrictConcaveOn.ae_eq_const_or_lt_map_average
+-/
 
+#print ae_eq_const_or_norm_average_lt_of_norm_le_const /-
 /-- If `E` is a strictly convex normed space and `f : α → E` is a function such that `‖f x‖ ≤ C`
 a.e., then either this function is a.e. equal to its average value, or the norm of its average value
 is strictly less than `C`. -/
@@ -354,7 +393,9 @@ theorem ae_eq_const_or_norm_average_lt_of_norm_le_const [StrictConvexSpace ℝ E
     (strictConvex_closedBall ℝ (0 : E) C).ae_eq_const_or_average_mem_interior is_closed_ball h_le
       hfi
 #align ae_eq_const_or_norm_average_lt_of_norm_le_const ae_eq_const_or_norm_average_lt_of_norm_le_const
+-/
 
+#print ae_eq_const_or_norm_integral_lt_of_norm_le_const /-
 /-- If `E` is a strictly convex normed space and `f : α → E` is a function such that `‖f x‖ ≤ C`
 a.e., then either this function is a.e. equal to its average value, or the norm of its integral is
 strictly less than `(μ univ).to_real * C`. -/
@@ -369,7 +410,9 @@ theorem ae_eq_const_or_norm_integral_lt_of_norm_le_const [StrictConvexSpace ℝ
   rwa [average_eq, norm_smul, norm_inv, Real.norm_eq_abs, abs_of_pos hμ, ← div_eq_inv_mul,
     div_lt_iff' hμ] at H 
 #align ae_eq_const_or_norm_integral_lt_of_norm_le_const ae_eq_const_or_norm_integral_lt_of_norm_le_const
+-/
 
+#print ae_eq_const_or_norm_set_integral_lt_of_norm_le_const /-
 /-- If `E` is a strictly convex normed space and `f : α → E` is a function such that `‖f x‖ ≤ C`
 a.e. on a set `t` of finite measure, then either this function is a.e. equal to its average value on
 `t`, or the norm of its integral over `t` is strictly less than `(μ t).to_real * C`. -/
@@ -381,4 +424,5 @@ theorem ae_eq_const_or_norm_set_integral_lt_of_norm_le_const [StrictConvexSpace
   rw [← restrict_apply_univ]
   exact ae_eq_const_or_norm_integral_lt_of_norm_le_const h_le
 #align ae_eq_const_or_norm_set_integral_lt_of_norm_le_const ae_eq_const_or_norm_set_integral_lt_of_norm_le_const
+-/
 

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

@@ -60,7 +60,7 @@ variable {α E F : Type _} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [Nor
 integrable function sending `μ`-a.e. points to `s`, then the expected value of `f` belongs to `s`:
 `∫ x, f x ∂μ ∈ s`. See also `convex.sum_mem` for a finite sum version of this lemma. -/
 theorem Convex.integral_mem [IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s)
-    (hf : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) : (∫ x, f x ∂μ) ∈ s :=
+    (hf : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) : ∫ x, f x ∂μ ∈ s :=
   by
   borelize E
   rcases hfi.ae_strongly_measurable with ⟨g, hgm, hfg⟩
@@ -94,7 +94,7 @@ theorem Convex.integral_mem [IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc :
 integrable function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `s`:
 `⨍ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem` for a finite sum version of this lemma. -/
 theorem Convex.average_mem [IsFiniteMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s) (hμ : μ ≠ 0)
-    (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) : (⨍ x, f x ∂μ) ∈ s :=
+    (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) : ⨍ x, f x ∂μ ∈ s :=
   by
   have : is_probability_measure ((μ univ)⁻¹ • μ) := is_probability_measure_smul hμ
   refine' hs.integral_mem hsc (ae_mono' _ hfs) hfi.to_average
@@ -105,7 +105,7 @@ theorem Convex.average_mem [IsFiniteMeasure μ] (hs : Convex ℝ s) (hsc : IsClo
 integrable function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `s`:
 `⨍ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem` for a finite sum version of this lemma. -/
 theorem Convex.set_average_mem (hs : Convex ℝ s) (hsc : IsClosed s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞)
-    (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s) (hfi : IntegrableOn f t μ) : (⨍ x in t, f x ∂μ) ∈ s :=
+    (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s) (hfi : IntegrableOn f t μ) : ⨍ x in t, f x ∂μ ∈ s :=
   by
   haveI : Fact (μ t < ∞) := ⟨ht.lt_top⟩
   refine' hs.average_mem hsc _ hfs hfi
@@ -116,8 +116,7 @@ theorem Convex.set_average_mem (hs : Convex ℝ s) (hsc : IsClosed s) (h0 : μ t
 function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `closure s`:
 `⨍ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem` for a finite sum version of this lemma. -/
 theorem Convex.set_average_mem_closure (hs : Convex ℝ s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞)
-    (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s) (hfi : IntegrableOn f t μ) :
-    (⨍ x in t, f x ∂μ) ∈ closure s :=
+    (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s) (hfi : IntegrableOn f t μ) : ⨍ x in t, f x ∂μ ∈ closure s :=
   hs.closure.set_average_mem isClosed_closure h0 ht (hfs.mono fun x hx => subset_closure hx) hfi
 #align convex.set_average_mem_closure Convex.set_average_mem_closure
 
@@ -158,7 +157,7 @@ at the average value of `f` provided that both `f` and `g ∘ f` are integrable.
 `concave_on.le_map_center_mass` for a finite sum version of this lemma. -/
 theorem ConcaveOn.le_map_average [IsFiniteMeasure μ] (hg : ConcaveOn ℝ s g) (hgc : ContinuousOn g s)
     (hsc : IsClosed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
-    (hgi : Integrable (g ∘ f) μ) : (⨍ x, g (f x) ∂μ) ≤ g (⨍ x, f x ∂μ) :=
+    (hgi : Integrable (g ∘ f) μ) : ⨍ x, g (f x) ∂μ ≤ g (⨍ x, f x ∂μ) :=
   (hg.average_mem_hypograph hgc hsc hμ hfs hfi hgi).2
 #align concave_on.le_map_average ConcaveOn.le_map_average
 
@@ -210,7 +209,7 @@ are integrable. -/
 theorem ConcaveOn.le_map_set_average (hg : ConcaveOn ℝ s g) (hgc : ContinuousOn g s)
     (hsc : IsClosed s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞) (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s)
     (hfi : IntegrableOn f t μ) (hgi : IntegrableOn (g ∘ f) t μ) :
-    (⨍ x in t, g (f x) ∂μ) ≤ g (⨍ x in t, f x ∂μ) :=
+    ⨍ x in t, g (f x) ∂μ ≤ g (⨍ x in t, f x ∂μ) :=
   (hg.set_average_mem_hypograph hgc hsc h0 ht hfs hfi hgi).2
 #align concave_on.le_map_set_average ConcaveOn.le_map_set_average
 
@@ -232,7 +231,7 @@ to `s`, then the expected value of `g ∘ f` is less than or equal to the value
 value of `f` provided that both `f` and `g ∘ f` are integrable. -/
 theorem ConcaveOn.le_map_integral [IsProbabilityMeasure μ] (hg : ConcaveOn ℝ s g)
     (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
-    (hgi : Integrable (g ∘ f) μ) : (∫ x, g (f x) ∂μ) ≤ g (∫ x, f x ∂μ) := by
+    (hgi : Integrable (g ∘ f) μ) : ∫ x, g (f x) ∂μ ≤ g (∫ x, f x ∂μ) := by
   simpa only [average_eq_integral] using
     hg.le_map_average hgc hsc (is_probability_measure.ne_zero μ) hfs hfi hgi
 #align concave_on.le_map_integral ConcaveOn.le_map_integral
@@ -247,7 +246,7 @@ theorem ConcaveOn.le_map_integral [IsProbabilityMeasure μ] (hg : ConcaveOn ℝ
 values of `f` over `t` and `tᶜ` are different. -/
 theorem ae_eq_const_or_exists_average_ne_compl [IsFiniteMeasure μ] (hfi : Integrable f μ) :
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨
-      ∃ t, MeasurableSet t ∧ μ t ≠ 0 ∧ μ (tᶜ) ≠ 0 ∧ (⨍ x in t, f x ∂μ) ≠ ⨍ x in tᶜ, f x ∂μ :=
+      ∃ t, MeasurableSet t ∧ μ t ≠ 0 ∧ μ (tᶜ) ≠ 0 ∧ ⨍ x in t, f x ∂μ ≠ ⨍ x in tᶜ, f x ∂μ :=
   by
   refine' or_iff_not_imp_right.mpr fun H => _; push_neg at H 
   refine' hfi.ae_eq_of_forall_set_integral_eq _ _ (integrable_const _) fun t ht ht' => _; clear ht'
@@ -266,8 +265,8 @@ theorem ae_eq_const_or_exists_average_ne_compl [IsFiniteMeasure μ] (hfi : Integ
 positive measure, the average value of `f` over `t` belongs to the interior of `s`, then the average
 of `f` over the whole space belongs to the interior of `s`. -/
 theorem Convex.average_mem_interior_of_set [IsFiniteMeasure μ] (hs : Convex ℝ s) (h0 : μ t ≠ 0)
-    (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) (ht : (⨍ x in t, f x ∂μ) ∈ interior s) :
-    (⨍ x, f x ∂μ) ∈ interior s :=
+    (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) (ht : ⨍ x in t, f x ∂μ ∈ interior s) :
+    ⨍ x, f x ∂μ ∈ interior s :=
   by
   rw [← measure_to_measurable] at h0 ; rw [← restrict_to_measurable (measure_ne_top μ t)] at ht 
   by_cases h0' : μ (to_measurable μ tᶜ) = 0
@@ -286,9 +285,9 @@ either it is a.e. equal to its average value, or its average value belongs to th
 `s`. -/
 theorem StrictConvex.ae_eq_const_or_average_mem_interior [IsFiniteMeasure μ] (hs : StrictConvex ℝ s)
     (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) :
-    f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ (⨍ x, f x ∂μ) ∈ interior s :=
+    f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ ⨍ x, f x ∂μ ∈ interior s :=
   by
-  have : ∀ {t}, μ t ≠ 0 → (⨍ x in t, f x ∂μ) ∈ s := fun t ht =>
+  have : ∀ {t}, μ t ≠ 0 → ⨍ x in t, f x ∂μ ∈ s := fun t ht =>
     hs.convex.set_average_mem hsc ht (measure_ne_top _ _) (ae_restrict_of_ae hfs) hfi.integrable_on
   refine' (ae_eq_const_or_exists_average_ne_compl hfi).imp_right _
   rintro ⟨t, hm, h₀, h₀', hne⟩
@@ -305,7 +304,7 @@ theorem StrictConvexOn.ae_eq_const_or_map_average_lt [IsFiniteMeasure μ] (hg :
     (hgi : Integrable (g ∘ f) μ) :
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ g (⨍ x, f x ∂μ) < ⨍ x, g (f x) ∂μ :=
   by
-  have : ∀ {t}, μ t ≠ 0 → (⨍ x in t, f x ∂μ) ∈ s ∧ g (⨍ x in t, f x ∂μ) ≤ ⨍ x in t, g (f x) ∂μ :=
+  have : ∀ {t}, μ t ≠ 0 → ⨍ x in t, f x ∂μ ∈ s ∧ g (⨍ x in t, f x ∂μ) ≤ ⨍ x in t, g (f x) ∂μ :=
     fun t ht =>
     hg.convex_on.set_average_mem_epigraph hgc hsc ht (measure_ne_top _ _) (ae_restrict_of_ae hfs)
       hfi.integrable_on hgi.integrable_on
@@ -317,10 +316,10 @@ theorem StrictConvexOn.ae_eq_const_or_map_average_lt [IsFiniteMeasure μ] (hg :
     Prod.mk_add_mk, Prod.mk.inj_iff, (· ∘ ·)] at h_avg 
   rw [← h_avg.1, ← h_avg.2]
   calc
-    g ((a • ⨍ x in t, f x ∂μ) + b • ⨍ x in tᶜ, f x ∂μ) <
+    g (a • ⨍ x in t, f x ∂μ + b • ⨍ x in tᶜ, f x ∂μ) <
         a * g (⨍ x in t, f x ∂μ) + b * g (⨍ x in tᶜ, f x ∂μ) :=
       hg.2 (this h₀).1 (this h₀').1 hne ha hb hab
-    _ ≤ (a * ⨍ x in t, g (f x) ∂μ) + b * ⨍ x in tᶜ, g (f x) ∂μ :=
+    _ ≤ a * ⨍ x in t, g (f x) ∂μ + b * ⨍ x in tᶜ, g (f x) ∂μ :=
       add_le_add (mul_le_mul_of_nonneg_left (this h₀).2 ha.le)
         (mul_le_mul_of_nonneg_left (this h₀').2 hb.le)
 #align strict_convex_on.ae_eq_const_or_map_average_lt StrictConvexOn.ae_eq_const_or_map_average_lt
@@ -331,7 +330,7 @@ either `f` is a.e. equal to its average value, or `⨍ x, g (f x) ∂μ < g (⨍
 theorem StrictConcaveOn.ae_eq_const_or_lt_map_average [IsFiniteMeasure μ]
     (hg : StrictConcaveOn ℝ s g) (hgc : ContinuousOn g s) (hsc : IsClosed s)
     (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
-    f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ (⨍ x, g (f x) ∂μ) < g (⨍ x, f x ∂μ) := by
+    f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ ⨍ x, g (f x) ∂μ < g (⨍ x, f x ∂μ) := by
   simpa only [Pi.neg_apply, average_neg, neg_lt_neg_iff] using
     hg.neg.ae_eq_const_or_map_average_lt hgc.neg hsc hfs hfi hgi.neg
 #align strict_concave_on.ae_eq_const_or_lt_map_average StrictConcaveOn.ae_eq_const_or_lt_map_average

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

@@ -323,7 +323,6 @@ theorem StrictConvexOn.ae_eq_const_or_map_average_lt [IsFiniteMeasure μ] (hg :
     _ ≤ (a * ⨍ x in t, g (f x) ∂μ) + b * ⨍ x in tᶜ, g (f x) ∂μ :=
       add_le_add (mul_le_mul_of_nonneg_left (this h₀).2 ha.le)
         (mul_le_mul_of_nonneg_left (this h₀').2 hb.le)
-    
 #align strict_convex_on.ae_eq_const_or_map_average_lt StrictConvexOn.ae_eq_const_or_map_average_lt
 
 /-- **Jensen's inequality**, strict version: if an integrable function `f : α → E` takes values in a

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

@@ -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 analysis.convex.integral
-! 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.
 -/
@@ -16,6 +16,9 @@ import Mathbin.MeasureTheory.Integral.Average
 /-!
 # Jensen's inequality for integrals
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we prove several forms of Jensen's inequality for integrals.
 
 - for convex sets: `convex.average_mem`, `convex.set_average_mem`, `convex.integral_mem`;

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

@@ -56,7 +56,7 @@ variable {α E F : Type _} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [Nor
 /-- If `μ` is a probability measure on `α`, `s` is a convex closed set in `E`, and `f` is an
 integrable function sending `μ`-a.e. points to `s`, then the expected value of `f` belongs to `s`:
 `∫ x, f x ∂μ ∈ s`. See also `convex.sum_mem` for a finite sum version of this lemma. -/
-theorem Convex.integral_mem [ProbabilityMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s)
+theorem Convex.integral_mem [IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s)
     (hf : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) : (∫ x, f x ∂μ) ∈ s :=
   by
   borelize E
@@ -70,7 +70,7 @@ theorem Convex.integral_mem [ProbabilityMeasure μ] (hs : Convex ℝ s) (hsc : I
   rw [integral_congr_ae hfg]; rw [integrable_congr hfg] at hfi 
   have hg : ∀ᵐ x ∂μ, g x ∈ closure (range g ∩ s) :=
     by
-    filter_upwards [hfg.rw (fun x y => y ∈ s) hf]with x hx
+    filter_upwards [hfg.rw (fun x y => y ∈ s) hf] with x hx
     apply subset_closure
     exact ⟨mem_range_self _, hx⟩
   set G : ℕ → simple_func α E := simple_func.approx_on _ hgm.measurable (range g ∩ s) y₀ h₀
@@ -90,7 +90,7 @@ theorem Convex.integral_mem [ProbabilityMeasure μ] (hs : Convex ℝ s) (hsc : I
 /-- If `μ` is a non-zero finite measure on `α`, `s` is a convex closed set in `E`, and `f` is an
 integrable function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `s`:
 `⨍ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem` for a finite sum version of this lemma. -/
-theorem Convex.average_mem [FiniteMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s) (hμ : μ ≠ 0)
+theorem Convex.average_mem [IsFiniteMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s) (hμ : μ ≠ 0)
     (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) : (⨍ x, f x ∂μ) ∈ s :=
   by
   have : is_probability_measure ((μ univ)⁻¹ • μ) := is_probability_measure_smul hμ
@@ -118,21 +118,21 @@ theorem Convex.set_average_mem_closure (hs : Convex ℝ s) (h0 : μ t ≠ 0) (ht
   hs.closure.set_average_mem isClosed_closure h0 ht (hfs.mono fun x hx => subset_closure hx) hfi
 #align convex.set_average_mem_closure Convex.set_average_mem_closure
 
-theorem ConvexOn.average_mem_epigraph [FiniteMeasure μ] (hg : ConvexOn ℝ s g)
+theorem ConvexOn.average_mem_epigraph [IsFiniteMeasure μ] (hg : ConvexOn ℝ s g)
     (hgc : ContinuousOn g s) (hsc : IsClosed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
     (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
-    (⨍ x, f x ∂μ, ⨍ x, g (f x) ∂μ) ∈ { p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2 } :=
+    (⨍ x, f x ∂μ, ⨍ x, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2} :=
   by
-  have ht_mem : ∀ᵐ x ∂μ, (f x, g (f x)) ∈ { p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2 } :=
+  have ht_mem : ∀ᵐ x ∂μ, (f x, g (f x)) ∈ {p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2} :=
     hfs.mono fun x hx => ⟨hx, le_rfl⟩
   simpa only [average_pair hfi hgi] using
     hg.convex_epigraph.average_mem (hsc.epigraph hgc) hμ ht_mem (hfi.prod_mk hgi)
 #align convex_on.average_mem_epigraph ConvexOn.average_mem_epigraph
 
-theorem ConcaveOn.average_mem_hypograph [FiniteMeasure μ] (hg : ConcaveOn ℝ s g)
+theorem ConcaveOn.average_mem_hypograph [IsFiniteMeasure μ] (hg : ConcaveOn ℝ s g)
     (hgc : ContinuousOn g s) (hsc : IsClosed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
     (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
-    (⨍ x, f x ∂μ, ⨍ x, g (f x) ∂μ) ∈ { p : E × ℝ | p.1 ∈ s ∧ p.2 ≤ g p.1 } := by
+    (⨍ x, f x ∂μ, ⨍ x, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ p.2 ≤ g p.1} := by
   simpa only [mem_set_of_eq, Pi.neg_apply, average_neg, neg_le_neg_iff] using
     hg.neg.average_mem_epigraph hgc.neg hsc hμ hfs hfi hgi.neg
 #align concave_on.average_mem_hypograph ConcaveOn.average_mem_hypograph
@@ -142,7 +142,7 @@ set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a func
 `μ`-a.e. points to `s`, then the value of `g` at the average value of `f` is less than or equal to
 the average value of `g ∘ f` provided that both `f` and `g ∘ f` are integrable. See also
 `convex_on.map_center_mass_le` for a finite sum version of this lemma. -/
-theorem ConvexOn.map_average_le [FiniteMeasure μ] (hg : ConvexOn ℝ s g) (hgc : ContinuousOn g s)
+theorem ConvexOn.map_average_le [IsFiniteMeasure μ] (hg : ConvexOn ℝ s g) (hgc : ContinuousOn g s)
     (hsc : IsClosed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
     (hgi : Integrable (g ∘ f) μ) : g (⨍ x, f x ∂μ) ≤ ⨍ x, g (f x) ∂μ :=
   (hg.average_mem_epigraph hgc hsc hμ hfs hfi hgi).2
@@ -153,7 +153,7 @@ set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a func
 `μ`-a.e. points to `s`, then the average value of `g ∘ f` is less than or equal to the value of `g`
 at the average value of `f` provided that both `f` and `g ∘ f` are integrable. See also
 `concave_on.le_map_center_mass` for a finite sum version of this lemma. -/
-theorem ConcaveOn.le_map_average [FiniteMeasure μ] (hg : ConcaveOn ℝ s g) (hgc : ContinuousOn g s)
+theorem ConcaveOn.le_map_average [IsFiniteMeasure μ] (hg : ConcaveOn ℝ s g) (hgc : ContinuousOn g s)
     (hsc : IsClosed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
     (hgi : Integrable (g ∘ f) μ) : (⨍ x, g (f x) ∂μ) ≤ g (⨍ x, f x ∂μ) :=
   (hg.average_mem_hypograph hgc hsc hμ hfs hfi hgi).2
@@ -167,7 +167,7 @@ integrable. -/
 theorem ConvexOn.set_average_mem_epigraph (hg : ConvexOn ℝ s g) (hgc : ContinuousOn g s)
     (hsc : IsClosed s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞) (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s)
     (hfi : IntegrableOn f t μ) (hgi : IntegrableOn (g ∘ f) t μ) :
-    (⨍ x in t, f x ∂μ, ⨍ x in t, g (f x) ∂μ) ∈ { p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2 } :=
+    (⨍ x in t, f x ∂μ, ⨍ x in t, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2} :=
   by
   haveI : Fact (μ t < ∞) := ⟨ht.lt_top⟩
   refine' hg.average_mem_epigraph hgc hsc _ hfs hfi hgi
@@ -182,7 +182,7 @@ are integrable. -/
 theorem ConcaveOn.set_average_mem_hypograph (hg : ConcaveOn ℝ s g) (hgc : ContinuousOn g s)
     (hsc : IsClosed s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞) (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s)
     (hfi : IntegrableOn f t μ) (hgi : IntegrableOn (g ∘ f) t μ) :
-    (⨍ x in t, f x ∂μ, ⨍ x in t, g (f x) ∂μ) ∈ { p : E × ℝ | p.1 ∈ s ∧ p.2 ≤ g p.1 } := by
+    (⨍ x in t, f x ∂μ, ⨍ x in t, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ p.2 ≤ g p.1} := by
   simpa only [mem_set_of_eq, Pi.neg_apply, average_neg, neg_le_neg_iff] using
     hg.neg.set_average_mem_epigraph hgc.neg hsc h0 ht hfs hfi hgi.neg
 #align concave_on.set_average_mem_hypograph ConcaveOn.set_average_mem_hypograph
@@ -216,7 +216,7 @@ set `s`, `μ` is a probability measure on `α`, and `f : α → E` is a function
 to `s`, then the value of `g` at the expected value of `f` is less than or equal to the expected
 value of `g ∘ f` provided that both `f` and `g ∘ f` are integrable. See also
 `convex_on.map_center_mass_le` for a finite sum version of this lemma. -/
-theorem ConvexOn.map_integral_le [ProbabilityMeasure μ] (hg : ConvexOn ℝ s g)
+theorem ConvexOn.map_integral_le [IsProbabilityMeasure μ] (hg : ConvexOn ℝ s g)
     (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
     (hgi : Integrable (g ∘ f) μ) : g (∫ x, f x ∂μ) ≤ ∫ x, g (f x) ∂μ := by
   simpa only [average_eq_integral] using
@@ -227,7 +227,7 @@ theorem ConvexOn.map_integral_le [ProbabilityMeasure μ] (hg : ConvexOn ℝ s g)
 set `s`, `μ` is a probability measure on `α`, and `f : α → E` is a function sending `μ`-a.e.  points
 to `s`, then the expected value of `g ∘ f` is less than or equal to the value of `g` at the expected
 value of `f` provided that both `f` and `g ∘ f` are integrable. -/
-theorem ConcaveOn.le_map_integral [ProbabilityMeasure μ] (hg : ConcaveOn ℝ s g)
+theorem ConcaveOn.le_map_integral [IsProbabilityMeasure μ] (hg : ConcaveOn ℝ s g)
     (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
     (hgi : Integrable (g ∘ f) μ) : (∫ x, g (f x) ∂μ) ≤ g (∫ x, f x ∂μ) := by
   simpa only [average_eq_integral] using
@@ -242,11 +242,11 @@ theorem ConcaveOn.le_map_integral [ProbabilityMeasure μ] (hg : ConcaveOn ℝ s
 /-- If `f : α → E` is an integrable function, then either it is a.e. equal to the constant
 `⨍ x, f x ∂μ` or there exists a measurable set such that `μ t ≠ 0`, `μ tᶜ ≠ 0`, and the average
 values of `f` over `t` and `tᶜ` are different. -/
-theorem ae_eq_const_or_exists_average_ne_compl [FiniteMeasure μ] (hfi : Integrable f μ) :
+theorem ae_eq_const_or_exists_average_ne_compl [IsFiniteMeasure μ] (hfi : Integrable f μ) :
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨
       ∃ t, MeasurableSet t ∧ μ t ≠ 0 ∧ μ (tᶜ) ≠ 0 ∧ (⨍ x in t, f x ∂μ) ≠ ⨍ x in tᶜ, f x ∂μ :=
   by
-  refine' or_iff_not_imp_right.mpr fun H => _; push_neg  at H 
+  refine' or_iff_not_imp_right.mpr fun H => _; push_neg at H 
   refine' hfi.ae_eq_of_forall_set_integral_eq _ _ (integrable_const _) fun t ht ht' => _; clear ht'
   simp only [const_apply, set_integral_const]
   by_cases h₀ : μ t = 0
@@ -262,7 +262,7 @@ theorem ae_eq_const_or_exists_average_ne_compl [FiniteMeasure μ] (hfi : Integra
 /-- If an integrable function `f : α → E` takes values in a convex set `s` and for some set `t` of
 positive measure, the average value of `f` over `t` belongs to the interior of `s`, then the average
 of `f` over the whole space belongs to the interior of `s`. -/
-theorem Convex.average_mem_interior_of_set [FiniteMeasure μ] (hs : Convex ℝ s) (h0 : μ t ≠ 0)
+theorem Convex.average_mem_interior_of_set [IsFiniteMeasure μ] (hs : Convex ℝ s) (h0 : μ t ≠ 0)
     (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) (ht : (⨍ x in t, f x ∂μ) ∈ interior s) :
     (⨍ x, f x ∂μ) ∈ interior s :=
   by
@@ -281,7 +281,7 @@ theorem Convex.average_mem_interior_of_set [FiniteMeasure μ] (hs : Convex ℝ s
 /-- If an integrable function `f : α → E` takes values in a strictly convex closed set `s`, then
 either it is a.e. equal to its average value, or its average value belongs to the interior of
 `s`. -/
-theorem StrictConvex.ae_eq_const_or_average_mem_interior [FiniteMeasure μ] (hs : StrictConvex ℝ s)
+theorem StrictConvex.ae_eq_const_or_average_mem_interior [IsFiniteMeasure μ] (hs : StrictConvex ℝ s)
     (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) :
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ (⨍ x, f x ∂μ) ∈ interior s :=
   by
@@ -297,7 +297,7 @@ theorem StrictConvex.ae_eq_const_or_average_mem_interior [FiniteMeasure μ] (hs
 /-- **Jensen's inequality**, strict version: if an integrable function `f : α → E` takes values in a
 convex closed set `s`, and `g : E → ℝ` is continuous and strictly convex on `s`, then
 either `f` is a.e. equal to its average value, or `g (⨍ x, f x ∂μ) < ⨍ x, g (f x) ∂μ`. -/
-theorem StrictConvexOn.ae_eq_const_or_map_average_lt [FiniteMeasure μ] (hg : StrictConvexOn ℝ s g)
+theorem StrictConvexOn.ae_eq_const_or_map_average_lt [IsFiniteMeasure μ] (hg : StrictConvexOn ℝ s g)
     (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
     (hgi : Integrable (g ∘ f) μ) :
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ g (⨍ x, f x ∂μ) < ⨍ x, g (f x) ∂μ :=
@@ -326,9 +326,9 @@ theorem StrictConvexOn.ae_eq_const_or_map_average_lt [FiniteMeasure μ] (hg : St
 /-- **Jensen's inequality**, strict version: if an integrable function `f : α → E` takes values in a
 convex closed set `s`, and `g : E → ℝ` is continuous and strictly concave on `s`, then
 either `f` is a.e. equal to its average value, or `⨍ x, g (f x) ∂μ < g (⨍ x, f x ∂μ)`. -/
-theorem StrictConcaveOn.ae_eq_const_or_lt_map_average [FiniteMeasure μ] (hg : StrictConcaveOn ℝ s g)
-    (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
-    (hgi : Integrable (g ∘ f) μ) :
+theorem StrictConcaveOn.ae_eq_const_or_lt_map_average [IsFiniteMeasure μ]
+    (hg : StrictConcaveOn ℝ s g) (hgc : ContinuousOn g s) (hsc : IsClosed s)
+    (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ (⨍ x, g (f x) ∂μ) < g (⨍ x, f x ∂μ) := by
   simpa only [Pi.neg_apply, average_neg, neg_lt_neg_iff] using
     hg.neg.ae_eq_const_or_map_average_lt hgc.neg hsc hfs hfi hgi.neg
@@ -357,7 +357,7 @@ theorem ae_eq_const_or_norm_average_lt_of_norm_le_const [StrictConvexSpace ℝ E
 /-- If `E` is a strictly convex normed space and `f : α → E` is a function such that `‖f x‖ ≤ C`
 a.e., then either this function is a.e. equal to its average value, or the norm of its integral is
 strictly less than `(μ univ).to_real * C`. -/
-theorem ae_eq_const_or_norm_integral_lt_of_norm_le_const [StrictConvexSpace ℝ E] [FiniteMeasure μ]
+theorem ae_eq_const_or_norm_integral_lt_of_norm_le_const [StrictConvexSpace ℝ E] [IsFiniteMeasure μ]
     (h_le : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) :
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ ‖∫ x, f x ∂μ‖ < (μ univ).toReal * C :=
   by

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

@@ -67,7 +67,7 @@ theorem Convex.integral_mem [ProbabilityMeasure μ] (hs : Convex ℝ s) (hsc : I
     by
     rcases(hf.and hfg).exists with ⟨x₀, h₀⟩
     exact ⟨f x₀, by simp only [h₀.2, mem_range_self], h₀.1⟩
-  rw [integral_congr_ae hfg]; rw [integrable_congr hfg] at hfi
+  rw [integral_congr_ae hfg]; rw [integrable_congr hfg] at hfi 
   have hg : ∀ᵐ x ∂μ, g x ∈ closure (range g ∩ s) :=
     by
     filter_upwards [hfg.rw (fun x y => y ∈ s) hf]with x hx
@@ -246,16 +246,16 @@ theorem ae_eq_const_or_exists_average_ne_compl [FiniteMeasure μ] (hfi : Integra
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨
       ∃ t, MeasurableSet t ∧ μ t ≠ 0 ∧ μ (tᶜ) ≠ 0 ∧ (⨍ x in t, f x ∂μ) ≠ ⨍ x in tᶜ, f x ∂μ :=
   by
-  refine' or_iff_not_imp_right.mpr fun H => _; push_neg  at H
+  refine' or_iff_not_imp_right.mpr fun H => _; push_neg  at H 
   refine' hfi.ae_eq_of_forall_set_integral_eq _ _ (integrable_const _) fun t ht ht' => _; clear ht'
   simp only [const_apply, set_integral_const]
   by_cases h₀ : μ t = 0
   · rw [restrict_eq_zero.2 h₀, integral_zero_measure, h₀, ENNReal.zero_toReal, zero_smul]
   by_cases h₀' : μ (tᶜ) = 0
-  · rw [← ae_eq_univ] at h₀'
+  · rw [← ae_eq_univ] at h₀' 
     rw [restrict_congr_set h₀', restrict_univ, measure_congr h₀', measure_smul_average]
   have := average_mem_open_segment_compl_self ht.null_measurable_set h₀ h₀' hfi
-  rw [← H t ht h₀ h₀', openSegment_same, mem_singleton_iff] at this
+  rw [← H t ht h₀ h₀', openSegment_same, mem_singleton_iff] at this 
   rw [this, measure_smul_set_average _ (measure_ne_top μ _)]
 #align ae_eq_const_or_exists_average_ne_compl ae_eq_const_or_exists_average_ne_compl
 
@@ -266,10 +266,10 @@ theorem Convex.average_mem_interior_of_set [FiniteMeasure μ] (hs : Convex ℝ s
     (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) (ht : (⨍ x in t, f x ∂μ) ∈ interior s) :
     (⨍ x, f x ∂μ) ∈ interior s :=
   by
-  rw [← measure_to_measurable] at h0; rw [← restrict_to_measurable (measure_ne_top μ t)] at ht
+  rw [← measure_to_measurable] at h0 ; rw [← restrict_to_measurable (measure_ne_top μ t)] at ht 
   by_cases h0' : μ (to_measurable μ tᶜ) = 0
-  · rw [← ae_eq_univ] at h0'
-    rwa [restrict_congr_set h0', restrict_univ] at ht
+  · rw [← ae_eq_univ] at h0' 
+    rwa [restrict_congr_set h0', restrict_univ] at ht 
   exact
     hs.open_segment_interior_closure_subset_interior ht
       (hs.set_average_mem_closure h0' (measure_ne_top _ _) (ae_restrict_of_ae hfs)
@@ -311,7 +311,7 @@ theorem StrictConvexOn.ae_eq_const_or_map_average_lt [FiniteMeasure μ] (hg : St
   rcases average_mem_open_segment_compl_self hm.null_measurable_set h₀ h₀' (hfi.prod_mk hgi) with
     ⟨a, b, ha, hb, hab, h_avg⟩
   simp only [average_pair hfi hgi, average_pair hfi.integrable_on hgi.integrable_on, Prod.smul_mk,
-    Prod.mk_add_mk, Prod.mk.inj_iff, (· ∘ ·)] at h_avg
+    Prod.mk_add_mk, Prod.mk.inj_iff, (· ∘ ·)] at h_avg 
   rw [← h_avg.1, ← h_avg.2]
   calc
     g ((a • ⨍ x in t, f x ∂μ) + b • ⨍ x in tᶜ, f x ∂μ) <
@@ -366,7 +366,7 @@ theorem ae_eq_const_or_norm_integral_lt_of_norm_le_const [StrictConvexSpace ℝ
     simp [ENNReal.toReal_pos_iff, pos_iff_ne_zero, h₀, measure_lt_top]
   refine' (ae_eq_const_or_norm_average_lt_of_norm_le_const h_le).imp_right fun H => _
   rwa [average_eq, norm_smul, norm_inv, Real.norm_eq_abs, abs_of_pos hμ, ← div_eq_inv_mul,
-    div_lt_iff' hμ] at H
+    div_lt_iff' hμ] at H 
 #align ae_eq_const_or_norm_integral_lt_of_norm_le_const ae_eq_const_or_norm_integral_lt_of_norm_le_const
 
 /-- If `E` is a strictly convex normed space and `f : α → E` is a function such that `‖f x‖ ≤ C`

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

@@ -42,7 +42,7 @@ convex, integral, center mass, average value, Jensen's inequality
 
 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}

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

@@ -67,8 +67,7 @@ theorem Convex.integral_mem [ProbabilityMeasure μ] (hs : Convex ℝ s) (hsc : I
     by
     rcases(hf.and hfg).exists with ⟨x₀, h₀⟩
     exact ⟨f x₀, by simp only [h₀.2, mem_range_self], h₀.1⟩
-  rw [integral_congr_ae hfg]
-  rw [integrable_congr hfg] at hfi
+  rw [integral_congr_ae hfg]; rw [integrable_congr hfg] at hfi
   have hg : ∀ᵐ x ∂μ, g x ∈ closure (range g ∩ s) :=
     by
     filter_upwards [hfg.rw (fun x y => y ∈ s) hf]with x hx
@@ -362,9 +361,7 @@ theorem ae_eq_const_or_norm_integral_lt_of_norm_le_const [StrictConvexSpace ℝ
     (h_le : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) :
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ ‖∫ x, f x ∂μ‖ < (μ univ).toReal * C :=
   by
-  cases' eq_or_ne μ 0 with h₀ h₀
-  · left
-    simp [h₀]
+  cases' eq_or_ne μ 0 with h₀ h₀; · left; simp [h₀]
   have hμ : 0 < (μ univ).toReal := by
     simp [ENNReal.toReal_pos_iff, pos_iff_ne_zero, h₀, measure_lt_top]
   refine' (ae_eq_const_or_norm_average_lt_of_norm_le_const h_le).imp_right fun H => _

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

@@ -56,7 +56,7 @@ variable {α E F : Type _} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [Nor
 /-- If `μ` is a probability measure on `α`, `s` is a convex closed set in `E`, and `f` is an
 integrable function sending `μ`-a.e. points to `s`, then the expected value of `f` belongs to `s`:
 `∫ x, f x ∂μ ∈ s`. See also `convex.sum_mem` for a finite sum version of this lemma. -/
-theorem Convex.integral_mem [IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s)
+theorem Convex.integral_mem [ProbabilityMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s)
     (hf : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) : (∫ x, f x ∂μ) ∈ s :=
   by
   borelize E
@@ -91,7 +91,7 @@ theorem Convex.integral_mem [IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc :
 /-- If `μ` is a non-zero finite measure on `α`, `s` is a convex closed set in `E`, and `f` is an
 integrable function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `s`:
 `⨍ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem` for a finite sum version of this lemma. -/
-theorem Convex.average_mem [IsFiniteMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s) (hμ : μ ≠ 0)
+theorem Convex.average_mem [FiniteMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s) (hμ : μ ≠ 0)
     (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) : (⨍ x, f x ∂μ) ∈ s :=
   by
   have : is_probability_measure ((μ univ)⁻¹ • μ) := is_probability_measure_smul hμ
@@ -119,7 +119,7 @@ theorem Convex.set_average_mem_closure (hs : Convex ℝ s) (h0 : μ t ≠ 0) (ht
   hs.closure.set_average_mem isClosed_closure h0 ht (hfs.mono fun x hx => subset_closure hx) hfi
 #align convex.set_average_mem_closure Convex.set_average_mem_closure
 
-theorem ConvexOn.average_mem_epigraph [IsFiniteMeasure μ] (hg : ConvexOn ℝ s g)
+theorem ConvexOn.average_mem_epigraph [FiniteMeasure μ] (hg : ConvexOn ℝ s g)
     (hgc : ContinuousOn g s) (hsc : IsClosed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
     (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
     (⨍ x, f x ∂μ, ⨍ x, g (f x) ∂μ) ∈ { p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2 } :=
@@ -130,7 +130,7 @@ theorem ConvexOn.average_mem_epigraph [IsFiniteMeasure μ] (hg : ConvexOn ℝ s
     hg.convex_epigraph.average_mem (hsc.epigraph hgc) hμ ht_mem (hfi.prod_mk hgi)
 #align convex_on.average_mem_epigraph ConvexOn.average_mem_epigraph
 
-theorem ConcaveOn.average_mem_hypograph [IsFiniteMeasure μ] (hg : ConcaveOn ℝ s g)
+theorem ConcaveOn.average_mem_hypograph [FiniteMeasure μ] (hg : ConcaveOn ℝ s g)
     (hgc : ContinuousOn g s) (hsc : IsClosed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
     (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
     (⨍ x, f x ∂μ, ⨍ x, g (f x) ∂μ) ∈ { p : E × ℝ | p.1 ∈ s ∧ p.2 ≤ g p.1 } := by
@@ -143,7 +143,7 @@ set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a func
 `μ`-a.e. points to `s`, then the value of `g` at the average value of `f` is less than or equal to
 the average value of `g ∘ f` provided that both `f` and `g ∘ f` are integrable. See also
 `convex_on.map_center_mass_le` for a finite sum version of this lemma. -/
-theorem ConvexOn.map_average_le [IsFiniteMeasure μ] (hg : ConvexOn ℝ s g) (hgc : ContinuousOn g s)
+theorem ConvexOn.map_average_le [FiniteMeasure μ] (hg : ConvexOn ℝ s g) (hgc : ContinuousOn g s)
     (hsc : IsClosed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
     (hgi : Integrable (g ∘ f) μ) : g (⨍ x, f x ∂μ) ≤ ⨍ x, g (f x) ∂μ :=
   (hg.average_mem_epigraph hgc hsc hμ hfs hfi hgi).2
@@ -154,7 +154,7 @@ set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a func
 `μ`-a.e. points to `s`, then the average value of `g ∘ f` is less than or equal to the value of `g`
 at the average value of `f` provided that both `f` and `g ∘ f` are integrable. See also
 `concave_on.le_map_center_mass` for a finite sum version of this lemma. -/
-theorem ConcaveOn.le_map_average [IsFiniteMeasure μ] (hg : ConcaveOn ℝ s g) (hgc : ContinuousOn g s)
+theorem ConcaveOn.le_map_average [FiniteMeasure μ] (hg : ConcaveOn ℝ s g) (hgc : ContinuousOn g s)
     (hsc : IsClosed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
     (hgi : Integrable (g ∘ f) μ) : (⨍ x, g (f x) ∂μ) ≤ g (⨍ x, f x ∂μ) :=
   (hg.average_mem_hypograph hgc hsc hμ hfs hfi hgi).2
@@ -217,7 +217,7 @@ set `s`, `μ` is a probability measure on `α`, and `f : α → E` is a function
 to `s`, then the value of `g` at the expected value of `f` is less than or equal to the expected
 value of `g ∘ f` provided that both `f` and `g ∘ f` are integrable. See also
 `convex_on.map_center_mass_le` for a finite sum version of this lemma. -/
-theorem ConvexOn.map_integral_le [IsProbabilityMeasure μ] (hg : ConvexOn ℝ s g)
+theorem ConvexOn.map_integral_le [ProbabilityMeasure μ] (hg : ConvexOn ℝ s g)
     (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
     (hgi : Integrable (g ∘ f) μ) : g (∫ x, f x ∂μ) ≤ ∫ x, g (f x) ∂μ := by
   simpa only [average_eq_integral] using
@@ -228,7 +228,7 @@ theorem ConvexOn.map_integral_le [IsProbabilityMeasure μ] (hg : ConvexOn ℝ s
 set `s`, `μ` is a probability measure on `α`, and `f : α → E` is a function sending `μ`-a.e.  points
 to `s`, then the expected value of `g ∘ f` is less than or equal to the value of `g` at the expected
 value of `f` provided that both `f` and `g ∘ f` are integrable. -/
-theorem ConcaveOn.le_map_integral [IsProbabilityMeasure μ] (hg : ConcaveOn ℝ s g)
+theorem ConcaveOn.le_map_integral [ProbabilityMeasure μ] (hg : ConcaveOn ℝ s g)
     (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
     (hgi : Integrable (g ∘ f) μ) : (∫ x, g (f x) ∂μ) ≤ g (∫ x, f x ∂μ) := by
   simpa only [average_eq_integral] using
@@ -243,7 +243,7 @@ theorem ConcaveOn.le_map_integral [IsProbabilityMeasure μ] (hg : ConcaveOn ℝ
 /-- If `f : α → E` is an integrable function, then either it is a.e. equal to the constant
 `⨍ x, f x ∂μ` or there exists a measurable set such that `μ t ≠ 0`, `μ tᶜ ≠ 0`, and the average
 values of `f` over `t` and `tᶜ` are different. -/
-theorem ae_eq_const_or_exists_average_ne_compl [IsFiniteMeasure μ] (hfi : Integrable f μ) :
+theorem ae_eq_const_or_exists_average_ne_compl [FiniteMeasure μ] (hfi : Integrable f μ) :
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨
       ∃ t, MeasurableSet t ∧ μ t ≠ 0 ∧ μ (tᶜ) ≠ 0 ∧ (⨍ x in t, f x ∂μ) ≠ ⨍ x in tᶜ, f x ∂μ :=
   by
@@ -263,7 +263,7 @@ theorem ae_eq_const_or_exists_average_ne_compl [IsFiniteMeasure μ] (hfi : Integ
 /-- If an integrable function `f : α → E` takes values in a convex set `s` and for some set `t` of
 positive measure, the average value of `f` over `t` belongs to the interior of `s`, then the average
 of `f` over the whole space belongs to the interior of `s`. -/
-theorem Convex.average_mem_interior_of_set [IsFiniteMeasure μ] (hs : Convex ℝ s) (h0 : μ t ≠ 0)
+theorem Convex.average_mem_interior_of_set [FiniteMeasure μ] (hs : Convex ℝ s) (h0 : μ t ≠ 0)
     (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) (ht : (⨍ x in t, f x ∂μ) ∈ interior s) :
     (⨍ x, f x ∂μ) ∈ interior s :=
   by
@@ -282,7 +282,7 @@ theorem Convex.average_mem_interior_of_set [IsFiniteMeasure μ] (hs : Convex ℝ
 /-- If an integrable function `f : α → E` takes values in a strictly convex closed set `s`, then
 either it is a.e. equal to its average value, or its average value belongs to the interior of
 `s`. -/
-theorem StrictConvex.ae_eq_const_or_average_mem_interior [IsFiniteMeasure μ] (hs : StrictConvex ℝ s)
+theorem StrictConvex.ae_eq_const_or_average_mem_interior [FiniteMeasure μ] (hs : StrictConvex ℝ s)
     (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) :
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ (⨍ x, f x ∂μ) ∈ interior s :=
   by
@@ -298,7 +298,7 @@ theorem StrictConvex.ae_eq_const_or_average_mem_interior [IsFiniteMeasure μ] (h
 /-- **Jensen's inequality**, strict version: if an integrable function `f : α → E` takes values in a
 convex closed set `s`, and `g : E → ℝ` is continuous and strictly convex on `s`, then
 either `f` is a.e. equal to its average value, or `g (⨍ x, f x ∂μ) < ⨍ x, g (f x) ∂μ`. -/
-theorem StrictConvexOn.ae_eq_const_or_map_average_lt [IsFiniteMeasure μ] (hg : StrictConvexOn ℝ s g)
+theorem StrictConvexOn.ae_eq_const_or_map_average_lt [FiniteMeasure μ] (hg : StrictConvexOn ℝ s g)
     (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
     (hgi : Integrable (g ∘ f) μ) :
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ g (⨍ x, f x ∂μ) < ⨍ x, g (f x) ∂μ :=
@@ -327,9 +327,9 @@ theorem StrictConvexOn.ae_eq_const_or_map_average_lt [IsFiniteMeasure μ] (hg :
 /-- **Jensen's inequality**, strict version: if an integrable function `f : α → E` takes values in a
 convex closed set `s`, and `g : E → ℝ` is continuous and strictly concave on `s`, then
 either `f` is a.e. equal to its average value, or `⨍ x, g (f x) ∂μ < g (⨍ x, f x ∂μ)`. -/
-theorem StrictConcaveOn.ae_eq_const_or_lt_map_average [IsFiniteMeasure μ]
-    (hg : StrictConcaveOn ℝ s g) (hgc : ContinuousOn g s) (hsc : IsClosed s)
-    (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
+theorem StrictConcaveOn.ae_eq_const_or_lt_map_average [FiniteMeasure μ] (hg : StrictConcaveOn ℝ s g)
+    (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
+    (hgi : Integrable (g ∘ f) μ) :
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ (⨍ x, g (f x) ∂μ) < g (⨍ x, f x ∂μ) := by
   simpa only [Pi.neg_apply, average_neg, neg_lt_neg_iff] using
     hg.neg.ae_eq_const_or_map_average_lt hgc.neg hsc hfs hfi hgi.neg
@@ -358,7 +358,7 @@ theorem ae_eq_const_or_norm_average_lt_of_norm_le_const [StrictConvexSpace ℝ E
 /-- If `E` is a strictly convex normed space and `f : α → E` is a function such that `‖f x‖ ≤ C`
 a.e., then either this function is a.e. equal to its average value, or the norm of its integral is
 strictly less than `(μ univ).to_real * C`. -/
-theorem ae_eq_const_or_norm_integral_lt_of_norm_le_const [StrictConvexSpace ℝ E] [IsFiniteMeasure μ]
+theorem ae_eq_const_or_norm_integral_lt_of_norm_le_const [StrictConvexSpace ℝ E] [FiniteMeasure μ]
     (h_le : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) :
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ ‖∫ x, f x ∂μ‖ < (μ univ).toReal * C :=
   by

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

@@ -42,7 +42,7 @@ convex, integral, center mass, average value, Jensen's inequality
 
 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}
@@ -78,9 +78,9 @@ theorem Convex.integral_mem [IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc :
   have : tendsto (fun n => (G n).integral μ) at_top (𝓝 <| ∫ x, g x ∂μ) :=
     tendsto_integral_approx_on_of_measurable hfi _ hg _ (integrable_const _)
   refine' hsc.mem_of_tendsto this (eventually_of_forall fun n => hs.sum_mem _ _ _)
-  · exact fun _ _ => Ennreal.toReal_nonneg
-  · rw [← Ennreal.toReal_sum, (G n).sum_range_measure_preimage_singleton, measure_univ,
-      Ennreal.one_toReal]
+  · exact fun _ _ => ENNReal.toReal_nonneg
+  · rw [← ENNReal.toReal_sum, (G n).sum_range_measure_preimage_singleton, measure_univ,
+      ENNReal.one_toReal]
     exact fun _ _ => measure_ne_top _ _
   · simp only [simple_func.mem_range, forall_range_iff]
     intro x
@@ -251,7 +251,7 @@ theorem ae_eq_const_or_exists_average_ne_compl [IsFiniteMeasure μ] (hfi : Integ
   refine' hfi.ae_eq_of_forall_set_integral_eq _ _ (integrable_const _) fun t ht ht' => _; clear ht'
   simp only [const_apply, set_integral_const]
   by_cases h₀ : μ t = 0
-  · rw [restrict_eq_zero.2 h₀, integral_zero_measure, h₀, Ennreal.zero_toReal, zero_smul]
+  · rw [restrict_eq_zero.2 h₀, integral_zero_measure, h₀, ENNReal.zero_toReal, zero_smul]
   by_cases h₀' : μ (tᶜ) = 0
   · rw [← ae_eq_univ] at h₀'
     rw [restrict_congr_set h₀', restrict_univ, measure_congr h₀', measure_smul_average]
@@ -346,7 +346,7 @@ theorem ae_eq_const_or_norm_average_lt_of_norm_le_const [StrictConvexSpace ℝ E
     simp only [average_congr this, Pi.zero_apply, average_zero]
     exact Or.inl this
   by_cases hfi : integrable f μ; swap
-  · simp [average_eq, integral_undef hfi, hC0, Ennreal.toReal_pos_iff]
+  · simp [average_eq, integral_undef hfi, hC0, ENNReal.toReal_pos_iff]
   cases' (le_top : μ univ ≤ ∞).eq_or_lt with hμt hμt; · simp [average_eq, hμt, hC0]
   haveI : is_finite_measure μ := ⟨hμt⟩
   replace h_le : ∀ᵐ x ∂μ, f x ∈ closed_ball (0 : E) C; · simpa only [mem_closedBall_zero_iff]
@@ -366,7 +366,7 @@ theorem ae_eq_const_or_norm_integral_lt_of_norm_le_const [StrictConvexSpace ℝ
   · left
     simp [h₀]
   have hμ : 0 < (μ univ).toReal := by
-    simp [Ennreal.toReal_pos_iff, pos_iff_ne_zero, h₀, measure_lt_top]
+    simp [ENNReal.toReal_pos_iff, pos_iff_ne_zero, h₀, measure_lt_top]
   refine' (ae_eq_const_or_norm_average_lt_of_norm_le_const h_le).imp_right fun H => _
   rwa [average_eq, norm_smul, norm_inv, Real.norm_eq_abs, abs_of_pos hμ, ← div_eq_inv_mul,
     div_lt_iff' hμ] at H

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

Occasionally, remove a "deprecated by" or "deprecated since", to fit the line length.

This is desirable (to me) because

• it's more compact: I don't see a good reason for these declarations taking up more space than needed; as I understand it, deprecated lemmas are not supposed to be used in mathlib anyway
• putting the date on the same line as the attribute makes it easier to discover un-dated deprecations; they also ease writing a tool to replace these by a machine-readable version using leanprover/lean4#3968

@@ -364,6 +364,6 @@ theorem ae_eq_const_or_norm_setIntegral_lt_of_norm_le_const [StrictConvexSpace 
   exact ae_eq_const_or_norm_integral_lt_of_norm_le_const h_le
 #align ae_eq_const_or_norm_set_integral_lt_of_norm_le_const ae_eq_const_or_norm_setIntegral_lt_of_norm_le_const
 
-@[deprecated]
+@[deprecated] -- 2024-04-17
 alias ae_eq_const_or_norm_set_integral_lt_of_norm_le_const :=
-  ae_eq_const_or_norm_setIntegral_lt_of_norm_le_const -- deprecated on 2024-04-17
+  ae_eq_const_or_norm_setIntegral_lt_of_norm_le_const

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

@@ -235,8 +235,8 @@ theorem ae_eq_const_or_exists_average_ne_compl [IsFiniteMeasure μ] (hfi : Integ
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨
       ∃ t, MeasurableSet t ∧ μ t ≠ 0 ∧ μ tᶜ ≠ 0 ∧ (⨍ x in t, f x ∂μ) ≠ ⨍ x in tᶜ, f x ∂μ := by
   refine' or_iff_not_imp_right.mpr fun H => _; push_neg at H
-  refine' hfi.ae_eq_of_forall_set_integral_eq _ _ (integrable_const _) fun t ht ht' => _; clear ht'
-  simp only [const_apply, set_integral_const]
+  refine' hfi.ae_eq_of_forall_setIntegral_eq _ _ (integrable_const _) fun t ht ht' => _; clear ht'
+  simp only [const_apply, setIntegral_const]
   by_cases h₀ : μ t = 0
   · rw [restrict_eq_zero.2 h₀, integral_zero_measure, h₀, ENNReal.zero_toReal, zero_smul]
   by_cases h₀' : μ tᶜ = 0
@@ -356,10 +356,14 @@ theorem ae_eq_const_or_norm_integral_lt_of_norm_le_const [StrictConvexSpace ℝ
 /-- If `E` is a strictly convex normed space and `f : α → E` is a function such that `‖f x‖ ≤ C`
 a.e. on a set `t` of finite measure, then either this function is a.e. equal to its average value on
 `t`, or the norm of its integral over `t` is strictly less than `(μ t).toReal * C`. -/
-theorem ae_eq_const_or_norm_set_integral_lt_of_norm_le_const [StrictConvexSpace ℝ E] (ht : μ t ≠ ∞)
+theorem ae_eq_const_or_norm_setIntegral_lt_of_norm_le_const [StrictConvexSpace ℝ E] (ht : μ t ≠ ∞)
     (h_le : ∀ᵐ x ∂μ.restrict t, ‖f x‖ ≤ C) :
     f =ᵐ[μ.restrict t] const α (⨍ x in t, f x ∂μ) ∨ ‖∫ x in t, f x ∂μ‖ < (μ t).toReal * C := by
   haveI := Fact.mk ht.lt_top
   rw [← restrict_apply_univ]
   exact ae_eq_const_or_norm_integral_lt_of_norm_le_const h_le
-#align ae_eq_const_or_norm_set_integral_lt_of_norm_le_const ae_eq_const_or_norm_set_integral_lt_of_norm_le_const
+#align ae_eq_const_or_norm_set_integral_lt_of_norm_le_const ae_eq_const_or_norm_setIntegral_lt_of_norm_le_const
+
+@[deprecated]
+alias ae_eq_const_or_norm_set_integral_lt_of_norm_le_const :=
+  ae_eq_const_or_norm_setIntegral_lt_of_norm_le_const -- deprecated on 2024-04-17

ball for "bounded forall" and bex for "bounded exists" are from experience very confusing abbreviations. This PR renames them to forall_mem and exists_mem in the few Set lemma names that mention them.

Also deprecate ball_image_of_ball, mem_image_elim, mem_image_elim_on since those lemmas are duplicates of the renamed lemmas (apart from argument order and implicitness, which I am also fixing by making the binder in the RHS of forall_mem_image semi-implicit), have obscure names and are completely unused.

@@ -75,7 +75,7 @@ theorem Convex.integral_mem [IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc :
   · rw [← ENNReal.toReal_sum, (G n).sum_range_measure_preimage_singleton, measure_univ,
       ENNReal.one_toReal]
     exact fun _ _ => measure_ne_top _ _
-  · simp only [SimpleFunc.mem_range, forall_range_iff]
+  · simp only [SimpleFunc.mem_range, forall_mem_range]
     intro x
     apply inter_subset_right (range g)
     exact SimpleFunc.approxOn_mem hgm.measurable h₀ _ _

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>

@@ -333,7 +333,7 @@ theorem ae_eq_const_or_norm_average_lt_of_norm_le_const [StrictConvexSpace ℝ E
   · simp [average_eq, integral_undef hfi, hC0, ENNReal.toReal_pos_iff]
   rcases (le_top : μ univ ≤ ∞).eq_or_lt with hμt | hμt; · simp [average_eq, hμt, hC0]
   haveI : IsFiniteMeasure μ := ⟨hμt⟩
-  replace h_le : ∀ᵐ x ∂μ, f x ∈ closedBall (0 : E) C; · simpa only [mem_closedBall_zero_iff]
+  replace h_le : ∀ᵐ x ∂μ, f x ∈ closedBall (0 : E) C := by simpa only [mem_closedBall_zero_iff]
   simpa only [interior_closedBall _ hC0.ne', mem_ball_zero_iff] using
     (strictConvex_closedBall ℝ (0 : E) C).ae_eq_const_or_average_mem_interior isClosed_ball h_le
       hfi

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.

@@ -325,13 +325,13 @@ a.e., then either this function is a.e. equal to its average value, or the norm
 is strictly less than `C`. -/
 theorem ae_eq_const_or_norm_average_lt_of_norm_le_const [StrictConvexSpace ℝ E]
     (h_le : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) : f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ ‖⨍ x, f x ∂μ‖ < C := by
-  cases' le_or_lt C 0 with hC0 hC0
+  rcases le_or_lt C 0 with hC0 | hC0
   · have : f =ᵐ[μ] 0 := h_le.mono fun x hx => norm_le_zero_iff.1 (hx.trans hC0)
     simp only [average_congr this, Pi.zero_apply, average_zero]
     exact Or.inl this
   by_cases hfi : Integrable f μ; swap
   · simp [average_eq, integral_undef hfi, hC0, ENNReal.toReal_pos_iff]
-  cases' (le_top : μ univ ≤ ∞).eq_or_lt with hμt hμt; · simp [average_eq, hμt, hC0]
+  rcases (le_top : μ univ ≤ ∞).eq_or_lt with hμt | hμt; · simp [average_eq, hμt, hC0]
   haveI : IsFiniteMeasure μ := ⟨hμt⟩
   replace h_le : ∀ᵐ x ∂μ, f x ∈ closedBall (0 : E) C; · simpa only [mem_closedBall_zero_iff]
   simpa only [interior_closedBall _ hC0.ne', mem_ball_zero_iff] using
@@ -345,7 +345,7 @@ strictly less than `(μ univ).toReal * C`. -/
 theorem ae_eq_const_or_norm_integral_lt_of_norm_le_const [StrictConvexSpace ℝ E] [IsFiniteMeasure μ]
     (h_le : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) :
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ ‖∫ x, f x ∂μ‖ < (μ univ).toReal * C := by
-  cases' eq_or_ne μ 0 with h₀ h₀; · left; simp [h₀, EventuallyEq]
+  rcases eq_or_ne μ 0 with h₀ | h₀; · left; simp [h₀, EventuallyEq]
   have hμ : 0 < (μ univ).toReal := by
     simp [ENNReal.toReal_pos_iff, pos_iff_ne_zero, h₀, measure_lt_top]
   refine' (ae_eq_const_or_norm_average_lt_of_norm_le_const h_le).imp_right fun H => _

Replace rcases( with rcases (. Same thing for convert( and congrm(. No other change.

@@ -60,7 +60,7 @@ theorem Convex.integral_mem [IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc :
   haveI : SeparableSpace (range g ∩ s : Set E) :=
     (hgm.isSeparable_range.mono (inter_subset_left _ _)).separableSpace
   obtain ⟨y₀, h₀⟩ : (range g ∩ s).Nonempty := by
-    rcases(hf.and hfg).exists with ⟨x₀, h₀⟩
+    rcases (hf.and hfg).exists with ⟨x₀, h₀⟩
     exact ⟨f x₀, by simp only [h₀.2, mem_range_self], h₀.1⟩
   rw [integral_congr_ae hfg]; rw [integrable_congr hfg] at hfi
   have hg : ∀ᵐ x ∂μ, g x ∈ closure (range g ∩ s) := by

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

This has nice performance benefits.

@@ -41,7 +41,7 @@ open MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Func
 
 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 : Measure α}
   {s : Set E} {t : Set α} {f : α → E} {g : E → ℝ} {C : ℝ}
 

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;

@@ -84,9 +84,8 @@ theorem Convex.integral_mem [IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc :
 /-- If `μ` is a non-zero finite measure on `α`, `s` is a convex closed set in `E`, and `f` is an
 integrable function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `s`:
 `⨍ x, f x ∂μ ∈ s`. See also `Convex.centerMass_mem` for a finite sum version of this lemma. -/
-theorem Convex.average_mem [IsFiniteMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s) (hμ : μ ≠ 0)
+theorem Convex.average_mem [IsFiniteMeasure μ] [NeZero μ] (hs : Convex ℝ s) (hsc : IsClosed s)
     (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) : (⨍ x, f x ∂μ) ∈ s := by
-  have : IsProbabilityMeasure ((μ univ)⁻¹ • μ) := isProbabilityMeasureSmul hμ
   refine' hs.integral_mem hsc (ae_mono' _ hfs) hfi.to_average
   exact AbsolutelyContinuous.smul (refl _) _
 #align convex.average_mem Convex.average_mem
@@ -95,10 +94,10 @@ theorem Convex.average_mem [IsFiniteMeasure μ] (hs : Convex ℝ s) (hsc : IsClo
 integrable function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `s`:
 `⨍ x, f x ∂μ ∈ s`. See also `Convex.centerMass_mem` for a finite sum version of this lemma. -/
 theorem Convex.set_average_mem (hs : Convex ℝ s) (hsc : IsClosed s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞)
-    (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s) (hfi : IntegrableOn f t μ) : (⨍ x in t, f x ∂μ) ∈ s := by
-  haveI : Fact (μ t < ∞) := ⟨ht.lt_top⟩
-  refine' hs.average_mem hsc _ hfs hfi
-  rwa [Ne.def, restrict_eq_zero]
+    (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s) (hfi : IntegrableOn f t μ) : (⨍ x in t, f x ∂μ) ∈ s :=
+  have := Fact.mk ht.lt_top
+  have := NeZero.mk h0
+  hs.average_mem hsc hfs hfi
 #align convex.set_average_mem Convex.set_average_mem
 
 /-- If `μ` is a non-zero finite measure on `α`, `s` is a convex set in `E`, and `f` is an integrable
@@ -110,22 +109,22 @@ theorem Convex.set_average_mem_closure (hs : Convex ℝ s) (h0 : μ t ≠ 0) (ht
   hs.closure.set_average_mem isClosed_closure h0 ht (hfs.mono fun _ hx => subset_closure hx) hfi
 #align convex.set_average_mem_closure Convex.set_average_mem_closure
 
-theorem ConvexOn.average_mem_epigraph [IsFiniteMeasure μ] (hg : ConvexOn ℝ s g)
-    (hgc : ContinuousOn g s) (hsc : IsClosed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
+theorem ConvexOn.average_mem_epigraph [IsFiniteMeasure μ] [NeZero μ] (hg : ConvexOn ℝ s g)
+    (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
     (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
     (⨍ x, f x ∂μ, ⨍ x, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2} := by
   have ht_mem : ∀ᵐ x ∂μ, (f x, g (f x)) ∈ {p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2} :=
     hfs.mono fun x hx => ⟨hx, le_rfl⟩
   exact average_pair hfi hgi ▸
-    hg.convex_epigraph.average_mem (hsc.epigraph hgc) hμ ht_mem (hfi.prod_mk hgi)
+    hg.convex_epigraph.average_mem (hsc.epigraph hgc) ht_mem (hfi.prod_mk hgi)
 #align convex_on.average_mem_epigraph ConvexOn.average_mem_epigraph
 
-theorem ConcaveOn.average_mem_hypograph [IsFiniteMeasure μ] (hg : ConcaveOn ℝ s g)
-    (hgc : ContinuousOn g s) (hsc : IsClosed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
+theorem ConcaveOn.average_mem_hypograph [IsFiniteMeasure μ] [NeZero μ] (hg : ConcaveOn ℝ s g)
+    (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
     (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
     (⨍ x, f x ∂μ, ⨍ x, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ p.2 ≤ g p.1} := by
   simpa only [mem_setOf_eq, Pi.neg_apply, average_neg, neg_le_neg_iff] using
-    hg.neg.average_mem_epigraph hgc.neg hsc hμ hfs hfi hgi.neg
+    hg.neg.average_mem_epigraph hgc.neg hsc hfs hfi hgi.neg
 #align concave_on.average_mem_hypograph ConcaveOn.average_mem_hypograph
 
 /-- **Jensen's inequality**: if a function `g : E → ℝ` is convex and continuous on a convex closed
@@ -133,10 +132,11 @@ set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a func
 `μ`-a.e. points to `s`, then the value of `g` at the average value of `f` is less than or equal to
 the average value of `g ∘ f` provided that both `f` and `g ∘ f` are integrable. See also
 `ConvexOn.map_centerMass_le` for a finite sum version of this lemma. -/
-theorem ConvexOn.map_average_le [IsFiniteMeasure μ] (hg : ConvexOn ℝ s g) (hgc : ContinuousOn g s)
-    (hsc : IsClosed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
-    (hgi : Integrable (g ∘ f) μ) : g (⨍ x, f x ∂μ) ≤ ⨍ x, g (f x) ∂μ :=
-  (hg.average_mem_epigraph hgc hsc hμ hfs hfi hgi).2
+theorem ConvexOn.map_average_le [IsFiniteMeasure μ] [NeZero μ]
+    (hg : ConvexOn ℝ s g) (hgc : ContinuousOn g s) (hsc : IsClosed s)
+    (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
+    g (⨍ x, f x ∂μ) ≤ ⨍ x, g (f x) ∂μ :=
+  (hg.average_mem_epigraph hgc hsc hfs hfi hgi).2
 #align convex_on.map_average_le ConvexOn.map_average_le
 
 /-- **Jensen's inequality**: if a function `g : E → ℝ` is concave and continuous on a convex closed
@@ -144,10 +144,11 @@ set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a func
 `μ`-a.e. points to `s`, then the average value of `g ∘ f` is less than or equal to the value of `g`
 at the average value of `f` provided that both `f` and `g ∘ f` are integrable. See also
 `ConcaveOn.le_map_centerMass` for a finite sum version of this lemma. -/
-theorem ConcaveOn.le_map_average [IsFiniteMeasure μ] (hg : ConcaveOn ℝ s g) (hgc : ContinuousOn g s)
-    (hsc : IsClosed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
-    (hgi : Integrable (g ∘ f) μ) : (⨍ x, g (f x) ∂μ) ≤ g (⨍ x, f x ∂μ) :=
-  (hg.average_mem_hypograph hgc hsc hμ hfs hfi hgi).2
+theorem ConcaveOn.le_map_average [IsFiniteMeasure μ] [NeZero μ]
+    (hg : ConcaveOn ℝ s g) (hgc : ContinuousOn g s) (hsc : IsClosed s)
+    (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
+    (⨍ x, g (f x) ∂μ) ≤ g (⨍ x, f x ∂μ) :=
+  (hg.average_mem_hypograph hgc hsc hfs hfi hgi).2
 #align concave_on.le_map_average ConcaveOn.le_map_average
 
 /-- **Jensen's inequality**: if a function `g : E → ℝ` is convex and continuous on a convex closed
@@ -158,10 +159,10 @@ integrable. -/
 theorem ConvexOn.set_average_mem_epigraph (hg : ConvexOn ℝ s g) (hgc : ContinuousOn g s)
     (hsc : IsClosed s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞) (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s)
     (hfi : IntegrableOn f t μ) (hgi : IntegrableOn (g ∘ f) t μ) :
-    (⨍ x in t, f x ∂μ, ⨍ x in t, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2} := by
-  haveI : Fact (μ t < ∞) := ⟨ht.lt_top⟩
-  refine' hg.average_mem_epigraph hgc hsc _ hfs hfi hgi
-  rwa [Ne.def, restrict_eq_zero]
+    (⨍ x in t, f x ∂μ, ⨍ x in t, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2} :=
+  have := Fact.mk ht.lt_top
+  have := NeZero.mk h0
+  hg.average_mem_epigraph hgc hsc hfs hfi hgi
 #align convex_on.set_average_mem_epigraph ConvexOn.set_average_mem_epigraph
 
 /-- **Jensen's inequality**: if a function `g : E → ℝ` is concave and continuous on a convex closed
@@ -209,8 +210,7 @@ value of `g ∘ f` provided that both `f` and `g ∘ f` are integrable. See also
 theorem ConvexOn.map_integral_le [IsProbabilityMeasure μ] (hg : ConvexOn ℝ s g)
     (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
     (hgi : Integrable (g ∘ f) μ) : g (∫ x, f x ∂μ) ≤ ∫ x, g (f x) ∂μ := by
-  simpa only [average_eq_integral] using
-    hg.map_average_le hgc hsc (IsProbabilityMeasure.ne_zero μ) hfs hfi hgi
+  simpa only [average_eq_integral] using hg.map_average_le hgc hsc hfs hfi hgi
 #align convex_on.map_integral_le ConvexOn.map_integral_le
 
 /-- **Jensen's inequality**: if a function `g : E → ℝ` is concave and continuous on a convex closed
@@ -220,8 +220,7 @@ value of `f` provided that both `f` and `g ∘ f` are integrable. -/
 theorem ConcaveOn.le_map_integral [IsProbabilityMeasure μ] (hg : ConcaveOn ℝ s g)
     (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
     (hgi : Integrable (g ∘ f) μ) : (∫ x, g (f x) ∂μ) ≤ g (∫ x, f x ∂μ) := by
-  simpa only [average_eq_integral] using
-    hg.le_map_average hgc hsc (IsProbabilityMeasure.ne_zero μ) hfs hfi hgi
+  simpa only [average_eq_integral] using hg.le_map_average hgc hsc hfs hfi hgi
 #align concave_on.le_map_integral ConcaveOn.le_map_integral
 
 /-!

• depends on: #5966

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

@@ -2,17 +2,14 @@
 Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
 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 analysis.convex.integral
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.Convex.Function
 import Mathlib.Analysis.Convex.StrictConvexSpace
 import Mathlib.MeasureTheory.Function.AEEqOfIntegral
 import Mathlib.MeasureTheory.Integral.Average
 
+#align_import analysis.convex.integral from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
+
 /-!
 # Jensen's inequality for integrals
 

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

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

@@ -248,7 +248,7 @@ theorem ae_eq_const_or_exists_average_ne_compl [IsFiniteMeasure μ] (hfi : Integ
     rw [restrict_congr_set h₀', restrict_univ, measure_congr h₀', measure_smul_average]
   have := average_mem_openSegment_compl_self ht.nullMeasurableSet h₀ h₀' hfi
   rw [← H t ht h₀ h₀', openSegment_same, mem_singleton_iff] at this
-  rw [this, measure_smul_set_average _ (measure_ne_top μ _)]
+  rw [this, measure_smul_setAverage _ (measure_ne_top μ _)]
 #align ae_eq_const_or_exists_average_ne_compl ae_eq_const_or_exists_average_ne_compl
 
 /-- If an integrable function `f : α → E` takes values in a convex set `s` and for some set `t` of

This is the second half of the changes originally in #5699, removing all occurrences of ; after a space and implementing a linter rule to enforce it.

In most cases this 2-character substring has a space after it, so the following command was run first:

find . -type f -name "*.lean" -exec sed -i -E 's/ ; /; /g' {} \;

The remaining cases were few enough in number that they were done manually.

@@ -257,7 +257,7 @@ of `f` over the whole space belongs to the interior of `s`. -/
 theorem Convex.average_mem_interior_of_set [IsFiniteMeasure μ] (hs : Convex ℝ s) (h0 : μ t ≠ 0)
     (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) (ht : (⨍ x in t, f x ∂μ) ∈ interior s) :
     (⨍ x, f x ∂μ) ∈ interior s := by
-  rw [← measure_toMeasurable] at h0 ; rw [← restrict_toMeasurable (measure_ne_top μ t)] at ht
+  rw [← measure_toMeasurable] at h0; rw [← restrict_toMeasurable (measure_ne_top μ t)] at ht
   by_cases h0' : μ (toMeasurable μ t)ᶜ = 0
   · rw [← ae_eq_univ] at h0'
     rwa [restrict_congr_set h0', restrict_univ] at ht

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

@@ -237,13 +237,13 @@ theorem ConcaveOn.le_map_integral [IsProbabilityMeasure μ] (hg : ConcaveOn ℝ
 values of `f` over `t` and `tᶜ` are different. -/
 theorem ae_eq_const_or_exists_average_ne_compl [IsFiniteMeasure μ] (hfi : Integrable f μ) :
     f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨
-      ∃ t, MeasurableSet t ∧ μ t ≠ 0 ∧ μ (tᶜ) ≠ 0 ∧ (⨍ x in t, f x ∂μ) ≠ ⨍ x in tᶜ, f x ∂μ := by
+      ∃ t, MeasurableSet t ∧ μ t ≠ 0 ∧ μ tᶜ ≠ 0 ∧ (⨍ x in t, f x ∂μ) ≠ ⨍ x in tᶜ, f x ∂μ := by
   refine' or_iff_not_imp_right.mpr fun H => _; push_neg at H
   refine' hfi.ae_eq_of_forall_set_integral_eq _ _ (integrable_const _) fun t ht ht' => _; clear ht'
   simp only [const_apply, set_integral_const]
   by_cases h₀ : μ t = 0
   · rw [restrict_eq_zero.2 h₀, integral_zero_measure, h₀, ENNReal.zero_toReal, zero_smul]
-  by_cases h₀' : μ (tᶜ) = 0
+  by_cases h₀' : μ tᶜ = 0
   · rw [← ae_eq_univ] at h₀'
     rw [restrict_congr_set h₀', restrict_univ, measure_congr h₀', measure_smul_average]
   have := average_mem_openSegment_compl_self ht.nullMeasurableSet h₀ h₀' hfi
@@ -258,7 +258,7 @@ theorem Convex.average_mem_interior_of_set [IsFiniteMeasure μ] (hs : Convex ℝ
     (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) (ht : (⨍ x in t, f x ∂μ) ∈ interior s) :
     (⨍ x, f x ∂μ) ∈ interior s := by
   rw [← measure_toMeasurable] at h0 ; rw [← restrict_toMeasurable (measure_ne_top μ t)] at ht
-  by_cases h0' : μ (toMeasurable μ tᶜ) = 0
+  by_cases h0' : μ (toMeasurable μ t)ᶜ = 0
   · rw [← ae_eq_univ] at h0'
     rwa [restrict_congr_set h0', restrict_univ] at ht
   exact

Co-authored-by: Komyyy <pol_tta@outlook.jp>