Lower Lebesgue integral for ℝ≥0∞-valued functions

We define the lower Lebesgue integral of an ℝ≥0∞-valued function.

Notation

We introduce the following notation for the lower Lebesgue integral of a function f : α → ℝ≥0∞.

• ∫⁻ x, f x ∂μ: integral of a function f : α → ℝ≥0∞ with respect to a measure μ;
• ∫⁻ x, f x: integral of a function f : α → ℝ≥0∞ with respect to the canonical measure volume on α;
• ∫⁻ x in s, f x ∂μ: integral of a function f : α → ℝ≥0∞ over a set s with respect to a measure μ, defined as ∫⁻ x, f x ∂(μ.restrict s);
• ∫⁻ x in s, f x: integral of a function f : α → ℝ≥0∞ over a set s with respect to the canonical measure volume, defined as ∫⁻ x, f x ∂(volume.restrict s).

theorem MeasureTheory.lintegral_def {α : Type u_5} : ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ENNReal), MeasureTheory.lintegral μ f = ⨆ (g : MeasureTheory.SimpleFunc α ENNReal) (_ : ↑g ≤ f), MeasureTheory.SimpleFunc.lintegral g μ

@[irreducible]

def MeasureTheory.lintegral {α : Type u_5} : {x : MeasurableSpace α} → MeasureTheory.Measure α → (α → ENNReal) → ENNReal

The lower Lebesgue integral of a function f with respect to a measure μ.

In the notation for integrals, an expression like ∫⁻ x, g ‖x‖ ∂μ will not be parsed correctly, and needs parentheses. We do not set the binding power of r to 0, because then ∫⁻ x, f x = 0 will be parsed incorrectly.

def MeasureTheory.«term∫⁻_,_∂_».delab : Lean.PrettyPrinter.Delaborator.Delab

def MeasureTheory.«term∫⁻_,_∂_» : Lean.ParserDescr

The lower Lebesgue integral of a function f with respect to a measure μ.

def MeasureTheory.«term∫⁻_,_».delab : Lean.PrettyPrinter.Delaborator.Delab

def MeasureTheory.«term∫⁻_,_» : Lean.ParserDescr

The lower Lebesgue integral of a function f with respect to a measure μ.

def MeasureTheory.«term∫⁻_In_,_∂_».delab : Lean.PrettyPrinter.Delaborator.Delab

def MeasureTheory.«term∫⁻_In_,_∂_» : Lean.ParserDescr

The lower Lebesgue integral of a function f with respect to a measure μ.

def MeasureTheory.«term∫⁻_In_,_» : Lean.ParserDescr

The lower Lebesgue integral of a function f with respect to a measure μ.

def MeasureTheory.«term∫⁻_In_,_».delab : Lean.PrettyPrinter.Delaborator.Delab

theorem MeasureTheory.SimpleFunc.lintegral_eq_lintegral {α : Type u_1} {m : MeasurableSpace α} (f : MeasureTheory.SimpleFunc α ENNReal) (μ : MeasureTheory.Measure α) : ∫⁻ (a : α), ↑f a ∂μ = MeasureTheory.SimpleFunc.lintegral f μ

theorem MeasureTheory.lintegral_mono' {α : Type u_1} {m : MeasurableSpace α} ⦃μ : MeasureTheory.Measure α⦄ ⦃ν : MeasureTheory.Measure α⦄ (hμν : μ ≤ ν) ⦃f : α → ENNReal⦄ ⦃g : α → ENNReal⦄ (hfg : f ≤ g) : ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), g a ∂ν

theorem MeasureTheory.lintegral_mono {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} ⦃f : α → ENNReal⦄ ⦃g : α → ENNReal⦄ (hfg : f ≤ g) : ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), g a ∂μ

theorem MeasureTheory.lintegral_mono_nnreal {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → NNReal} {g : α → NNReal} (h : f ≤ g) : ∫⁻ (a : α), ↑(f a) ∂μ ≤ ∫⁻ (a : α), ↑(g a) ∂μ

theorem MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ENNReal) : ⨆ (g : α → ENNReal) (_ : Measurable g) (_ : g ≤ f), ∫⁻ (a : α), g a ∂μ = ∫⁻ (a : α), f a ∂μ

theorem MeasureTheory.lintegral_mono_set {α : Type u_1} : ∀ {x : MeasurableSpace α} ⦃μ : MeasureTheory.Measure α⦄ {s t : Set α} {f : α → ENNReal}, s ⊆ t → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in t, f x ∂μ

theorem MeasureTheory.lintegral_mono_set' {α : Type u_1} : ∀ {x : MeasurableSpace α} ⦃μ : MeasureTheory.Measure α⦄ {s t : Set α} {f : α → ENNReal}, s ≤ᶠ[MeasureTheory.Measure.ae μ] t → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in t, f x ∂μ

theorem MeasureTheory.monotone_lintegral {α : Type u_1} : ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α), Monotone (MeasureTheory.lintegral μ)

@[simp]

theorem MeasureTheory.lintegral_const {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (c : ENNReal) : ∫⁻ (x : α), c ∂μ = c * ↑↑μ Set.univ

theorem MeasureTheory.lintegral_zero {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} : ∫⁻ (x : α), 0 ∂μ = 0

theorem MeasureTheory.lintegral_zero_fun {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} : MeasureTheory.lintegral μ 0 = 0

theorem MeasureTheory.lintegral_one {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} : ∫⁻ (x : α), 1 ∂μ = ↑↑μ Set.univ

theorem MeasureTheory.set_lintegral_const {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Set α) (c : ENNReal) : ∫⁻ (x : α) in s, c ∂μ = c * ↑↑μ s

theorem MeasureTheory.set_lintegral_one {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Set α) : ∫⁻ (x : α) in s, 1 ∂μ = ↑↑μ s

theorem MeasureTheory.set_lintegral_const_lt_top {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] (s : Set α) {c : ENNReal} (hc : c ≠ ⊤) : ∫⁻ (x : α) in s, c ∂μ < ⊤

theorem MeasureTheory.lintegral_const_lt_top {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {c : ENNReal} (hc : c ≠ ⊤) : ∫⁻ (x : α), c ∂μ < ⊤

theorem MeasureTheory.exists_measurable_le_lintegral_eq {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ENNReal) : ∃ g, Measurable g ∧ g ≤ f ∧ ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ

For any function f : α → ℝ≥0∞, there exists a measurable function g ≤ f with the same integral.

theorem MeasureTheory.lintegral_eq_nnreal {α : Type u_1} {m : MeasurableSpace α} (f : α → ENNReal) (μ : MeasureTheory.Measure α) : ∫⁻ (a : α), f a ∂μ = ⨆ (φ : MeasureTheory.SimpleFunc α NNReal) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), MeasureTheory.SimpleFunc.lintegral (MeasureTheory.SimpleFunc.map ENNReal.some φ) μ

∫⁻ a in s, f a ∂μ is defined as the supremum of integrals of simple functions φ : α →ₛ ℝ≥0∞ such that φ ≤ f. This lemma says that it suffices to take functions φ : α →ₛ ℝ≥0.

theorem MeasureTheory.exists_simpleFunc_forall_lintegral_sub_lt_of_pos {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (h : ∫⁻ (x : α), f x ∂μ ≠ ⊤) {ε : ENNReal} (hε : ε ≠ 0) : ∃ φ, (∀ (x : α), ↑(↑φ x) ≤ f x) ∧ ∀ (ψ : MeasureTheory.SimpleFunc α NNReal), (∀ (x : α), ↑(↑ψ x) ≤ f x) → MeasureTheory.SimpleFunc.lintegral (MeasureTheory.SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε

theorem MeasureTheory.iSup_lintegral_le {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Sort u_5} (f : ι → α → ENNReal) : ⨆ (i : ι), ∫⁻ (a : α), f i a ∂μ ≤ ∫⁻ (a : α), ⨆ (i : ι), f i a ∂μ

theorem MeasureTheory.iSup₂_lintegral_le {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Sort u_5} {ι' : ι → Sort u_6} (f : (i : ι) → ι' i → α → ENNReal) : ⨆ (i : ι) (j : ι' i), ∫⁻ (a : α), f i j a ∂μ ≤ ∫⁻ (a : α), ⨆ (i : ι) (j : ι' i), f i j a ∂μ

theorem MeasureTheory.le_iInf_lintegral {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Sort u_5} (f : ι → α → ENNReal) : ∫⁻ (a : α), ⨅ (i : ι), f i a ∂μ ≤ ⨅ (i : ι), ∫⁻ (a : α), f i a ∂μ

theorem MeasureTheory.le_iInf₂_lintegral {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Sort u_5} {ι' : ι → Sort u_6} (f : (i : ι) → ι' i → α → ENNReal) : ∫⁻ (a : α), ⨅ (i : ι) (h : ι' i), f i h a ∂μ ≤ ⨅ (i : ι) (h : ι' i), ∫⁻ (a : α), f i h a ∂μ

theorem MeasureTheory.lintegral_mono_ae {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (h : ∀ᵐ (a : α) ∂μ, f a ≤ g a) : ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), g a ∂μ

theorem MeasureTheory.set_lintegral_mono_ae {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ENNReal} {g : α → ENNReal} (hf : Measurable f) (hg : Measurable g) (hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → f x ≤ g x) : ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ

theorem MeasureTheory.set_lintegral_mono {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ENNReal} {g : α → ENNReal} (hf : Measurable f) (hg : Measurable g) (hfg : ∀ (x : α), x ∈ s → f x ≤ g x) : ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ

theorem MeasureTheory.set_lintegral_mono_ae' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ENNReal} {g : α → ENNReal} (hs : MeasurableSet s) (hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → f x ≤ g x) : ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ

theorem MeasureTheory.set_lintegral_mono' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ENNReal} {g : α → ENNReal} (hs : MeasurableSet s) (hfg : ∀ (x : α), x ∈ s → f x ≤ g x) : ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ

theorem MeasureTheory.lintegral_congr_ae {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (h : f =ᶠ[MeasureTheory.Measure.ae μ] g) : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ

theorem MeasureTheory.lintegral_congr {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (h : ∀ (a : α), f a = g a) : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ

theorem MeasureTheory.set_lintegral_congr {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {s : Set α} {t : Set α} (h : s =ᶠ[MeasureTheory.Measure.ae μ] t) : ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in t, f x ∂μ

theorem MeasureTheory.set_lintegral_congr_fun {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} {s : Set α} (hs : MeasurableSet s) (hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x) : ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ

theorem MeasureTheory.lintegral_ofReal_le_lintegral_nnnorm {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ) : ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ (x : α), ↑‖f x‖₊ ∂μ

theorem MeasureTheory.lintegral_nnnorm_eq_of_ae_nonneg {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (h_nonneg : 0 ≤ᶠ[MeasureTheory.Measure.ae μ] f) : ∫⁻ (x : α), ↑‖f x‖₊ ∂μ = ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ

theorem MeasureTheory.lintegral_nnnorm_eq_of_nonneg {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (h_nonneg : 0 ≤ f) : ∫⁻ (x : α), ↑‖f x‖₊ ∂μ = ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ

theorem MeasureTheory.lintegral_iSup {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ENNReal} (hf : ∀ (n : ℕ), Measurable (f n)) (h_mono : Monotone f) : ∫⁻ (a : α), ⨆ (n : ℕ), f n a ∂μ = ⨆ (n : ℕ), ∫⁻ (a : α), f n a ∂μ

Monotone convergence theorem -- sometimes called Beppo-Levi convergence. See lintegral_iSup_directed for a more general form.

theorem MeasureTheory.lintegral_iSup' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ENNReal} (hf : ∀ (n : ℕ), AEMeasurable (f n)) (h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n => f n x) : ∫⁻ (a : α), ⨆ (n : ℕ), f n a ∂μ = ⨆ (n : ℕ), ∫⁻ (a : α), f n a ∂μ

Monotone convergence theorem -- sometimes called Beppo-Levi convergence. Version with ae_measurable functions.

theorem MeasureTheory.lintegral_tendsto_of_tendsto_of_monotone {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ENNReal} {F : α → ENNReal} (hf : ∀ (n : ℕ), AEMeasurable (f n)) (h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n => f n x) (h_tendsto : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (nhds (F x))) : Filter.Tendsto (fun n => ∫⁻ (x : α), f n x ∂μ) Filter.atTop (nhds (∫⁻ (x : α), F x ∂μ))

Monotone convergence theorem expressed with limits

theorem MeasureTheory.lintegral_eq_iSup_eapprox_lintegral {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : Measurable f) : ∫⁻ (a : α), f a ∂μ = ⨆ (n : ℕ), MeasureTheory.SimpleFunc.lintegral (MeasureTheory.SimpleFunc.eapprox f n) μ

theorem MeasureTheory.exists_pos_set_lintegral_lt_of_measure_lt {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (h : ∫⁻ (x : α), f x ∂μ ≠ ⊤) {ε : ENNReal} (hε : ε ≠ 0) : ∃ δ, δ > 0 ∧ ∀ (s : Set α), ↑↑μ s < δ → ∫⁻ (x : α) in s, f x ∂μ < ε

If f has finite integral, then ∫⁻ x in s, f x ∂μ is absolutely continuous in s: it tends to zero as μ s tends to zero. This lemma states states this fact in terms of ε and δ.

theorem MeasureTheory.tendsto_set_lintegral_zero {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_5} {f : α → ENNReal} (h : ∫⁻ (x : α), f x ∂μ ≠ ⊤) {l : Filter ι} {s : ι → Set α} (hl : Filter.Tendsto (↑↑μ ∘ s) l (nhds 0)) : Filter.Tendsto (fun i => ∫⁻ (x : α) in s i, f x ∂μ) l (nhds 0)

If f has finite integral, then ∫⁻ x in s, f x ∂μ is absolutely continuous in s: it tends to zero as μ s tends to zero.

theorem MeasureTheory.le_lintegral_add {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ENNReal) (g : α → ENNReal) : ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ ≤ ∫⁻ (a : α), f a + g a ∂μ

The sum of the lower Lebesgue integrals of two functions is less than or equal to the integral of their sum. The other inequality needs one of these functions to be (a.e.-)measurable.

theorem MeasureTheory.lintegral_add_aux {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (hf : Measurable f) (hg : Measurable g) : ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ

@[simp]

theorem MeasureTheory.lintegral_add_left {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : Measurable f) (g : α → ENNReal) : ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ

If f g : α → ℝ≥0∞ are two functions and one of them is (a.e.) measurable, then the Lebesgue integral of f + g equals the sum of integrals. This lemma assumes that f is integrable, see also MeasureTheory.lintegral_add_right and primed versions of these lemmas.

theorem MeasureTheory.lintegral_add_left' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : AEMeasurable f) (g : α → ENNReal) : ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ

theorem MeasureTheory.lintegral_add_right' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ENNReal) {g : α → ENNReal} (hg : AEMeasurable g) : ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ

@[simp]

theorem MeasureTheory.lintegral_add_right {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ENNReal) {g : α → ENNReal} (hg : Measurable g) : ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ

If f g : α → ℝ≥0∞ are two functions and one of them is (a.e.) measurable, then the Lebesgue integral of f + g equals the sum of integrals. This lemma assumes that g is integrable, see also MeasureTheory.lintegral_add_left and primed versions of these lemmas.

@[simp]

theorem MeasureTheory.lintegral_smul_measure {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (c : ENNReal) (f : α → ENNReal) : ∫⁻ (a : α), f a ∂c • μ = c * ∫⁻ (a : α), f a ∂μ

@[simp]

theorem MeasureTheory.lintegral_sum_measure {α : Type u_1} {m : MeasurableSpace α} {ι : Type u_5} (f : α → ENNReal) (μ : ι → MeasureTheory.Measure α) : ∫⁻ (a : α), f a ∂MeasureTheory.Measure.sum μ = ∑' (i : ι), ∫⁻ (a : α), f a ∂μ i

theorem MeasureTheory.hasSum_lintegral_measure {α : Type u_1} {ι : Type u_5} : ∀ {x : MeasurableSpace α} (f : α → ENNReal) (μ : ι → MeasureTheory.Measure α), HasSum (fun i => ∫⁻ (a : α), f a ∂μ i) (∫⁻ (a : α), f a ∂MeasureTheory.Measure.sum μ)

@[simp]

theorem MeasureTheory.lintegral_add_measure {α : Type u_1} {m : MeasurableSpace α} (f : α → ENNReal) (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) : ∫⁻ (a : α), f a ∂(μ + ν) = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), f a ∂ν

@[simp]

theorem MeasureTheory.lintegral_finset_sum_measure {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} (s : Finset ι) (f : α → ENNReal) (μ : ι → MeasureTheory.Measure α) : (∫⁻ (a : α), f a ∂Finset.sum s fun i => μ i) = Finset.sum s fun i => ∫⁻ (a : α), f a ∂μ i

@[simp]

theorem MeasureTheory.lintegral_zero_measure {α : Type u_1} {m : MeasurableSpace α} (f : α → ENNReal) : ∫⁻ (a : α), f a ∂0 = 0

theorem MeasureTheory.set_lintegral_empty {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ENNReal) : ∫⁻ (x : α) in ∅, f x ∂μ = 0

theorem MeasureTheory.set_lintegral_univ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ENNReal) : ∫⁻ (x : α) in Set.univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ

theorem MeasureTheory.set_lintegral_measure_zero {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Set α) (f : α → ENNReal) (hs' : ↑↑μ s = 0) : ∫⁻ (x : α) in s, f x ∂μ = 0

theorem MeasureTheory.lintegral_finset_sum' {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Finset β) {f : β → α → ENNReal} (hf : ∀ (b : β), b ∈ s → AEMeasurable (f b)) : ∫⁻ (a : α), Finset.sum s fun b => f b a ∂μ = Finset.sum s fun b => ∫⁻ (a : α), f b a ∂μ

theorem MeasureTheory.lintegral_finset_sum {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Finset β) {f : β → α → ENNReal} (hf : ∀ (b : β), b ∈ s → Measurable (f b)) : ∫⁻ (a : α), Finset.sum s fun b => f b a ∂μ = Finset.sum s fun b => ∫⁻ (a : α), f b a ∂μ

@[simp]

theorem MeasureTheory.lintegral_const_mul {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ENNReal) {f : α → ENNReal} (hf : Measurable f) : ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ

theorem MeasureTheory.lintegral_const_mul'' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ENNReal) {f : α → ENNReal} (hf : AEMeasurable f) : ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ

theorem MeasureTheory.lintegral_const_mul_le {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ENNReal) (f : α → ENNReal) : r * ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), r * f a ∂μ

theorem MeasureTheory.lintegral_const_mul' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ENNReal) (f : α → ENNReal) (hr : r ≠ ⊤) : ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ

theorem MeasureTheory.lintegral_mul_const {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ENNReal) {f : α → ENNReal} (hf : Measurable f) : ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r

theorem MeasureTheory.lintegral_mul_const'' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ENNReal) {f : α → ENNReal} (hf : AEMeasurable f) : ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r

theorem MeasureTheory.lintegral_mul_const_le {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ENNReal) (f : α → ENNReal) : (∫⁻ (a : α), f a ∂μ) * r ≤ ∫⁻ (a : α), f a * r ∂μ

theorem MeasureTheory.lintegral_mul_const' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ENNReal) (f : α → ENNReal) (hr : r ≠ ⊤) : ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r

theorem MeasureTheory.lintegral_lintegral_mul {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [MeasurableSpace β] {ν : MeasureTheory.Measure β} {f : α → ENNReal} {g : β → ENNReal} (hf : AEMeasurable f) (hg : AEMeasurable g) : ∫⁻ (x : α), ∫⁻ (y : β), f x * g y ∂ν ∂μ = (∫⁻ (x : α), f x ∂μ) * ∫⁻ (y : β), g y ∂ν

theorem MeasureTheory.lintegral_rw₁ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → β} {f' : α → β} (h : f =ᶠ[MeasureTheory.Measure.ae μ] f') (g : β → ENNReal) : ∫⁻ (a : α), g (f a) ∂μ = ∫⁻ (a : α), g (f' a) ∂μ

theorem MeasureTheory.lintegral_rw₂ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f₁ : α → β} {f₁' : α → β} {f₂ : α → γ} {f₂' : α → γ} (h₁ : f₁ =ᶠ[MeasureTheory.Measure.ae μ] f₁') (h₂ : f₂ =ᶠ[MeasureTheory.Measure.ae μ] f₂') (g : β → γ → ENNReal) : ∫⁻ (a : α), g (f₁ a) (f₂ a) ∂μ = ∫⁻ (a : α), g (f₁' a) (f₂' a) ∂μ

@[simp]

theorem MeasureTheory.lintegral_indicator {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ENNReal) {s : Set α} (hs : MeasurableSet s) : ∫⁻ (a : α), Set.indicator s f a ∂μ = ∫⁻ (a : α) in s, f a ∂μ

theorem MeasureTheory.lintegral_indicator₀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ENNReal) {s : Set α} (hs : MeasureTheory.NullMeasurableSet s) : ∫⁻ (a : α), Set.indicator s f a ∂μ = ∫⁻ (a : α) in s, f a ∂μ

theorem MeasureTheory.lintegral_indicator_const₀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasureTheory.NullMeasurableSet s) (c : ENNReal) : ∫⁻ (a : α), Set.indicator s (fun x => c) a ∂μ = c * ↑↑μ s

theorem MeasureTheory.lintegral_indicator_const {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (c : ENNReal) : ∫⁻ (a : α), Set.indicator s (fun x => c) a ∂μ = c * ↑↑μ s

theorem MeasureTheory.set_lintegral_eq_const {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : Measurable f) (r : ENNReal) : ∫⁻ (x : α) in {x | f x = r}, f x ∂μ = r * ↑↑μ {x | f x = r}

@[simp]

theorem MeasureTheory.lintegral_indicator_one {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) : ∫⁻ (a : α), Set.indicator s 1 a ∂μ = ↑↑μ s

theorem MeasureTheory.lintegral_add_mul_meas_add_le_le_lintegral {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (hle : f ≤ᶠ[MeasureTheory.Measure.ae μ] g) (hg : AEMeasurable g) (ε : ENNReal) : ∫⁻ (a : α), f a ∂μ + ε * ↑↑μ {x | f x + ε ≤ g x} ≤ ∫⁻ (a : α), g a ∂μ

A version of Markov's inequality for two functions. It doesn't follow from the standard Markov's inequality because we only assume measurability of g, not f.

theorem MeasureTheory.mul_meas_ge_le_lintegral₀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : AEMeasurable f) (ε : ENNReal) : ε * ↑↑μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ

Markov's inequality also known as Chebyshev's first inequality.

theorem MeasureTheory.mul_meas_ge_le_lintegral {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : Measurable f) (ε : ENNReal) : ε * ↑↑μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ

Markov's inequality also known as Chebyshev's first inequality. For a version assuming AEMeasurable, see mul_meas_ge_le_lintegral₀.

theorem MeasureTheory.lintegral_eq_top_of_measure_eq_top_ne_zero {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : AEMeasurable f) (hμf : ↑↑μ {x | f x = ⊤} ≠ 0) : ∫⁻ (x : α), f x ∂μ = ⊤

theorem MeasureTheory.setLintegral_eq_top_of_measure_eq_top_ne_zero {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {s : Set α} (hf : AEMeasurable f) (hμf : ↑↑μ {x | x ∈ s ∧ f x = ⊤} ≠ 0) : ∫⁻ (x : α) in s, f x ∂μ = ⊤

theorem MeasureTheory.measure_eq_top_of_lintegral_ne_top {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : AEMeasurable f) (hμf : ∫⁻ (x : α), f x ∂μ ≠ ⊤) : ↑↑μ {x | f x = ⊤} = 0

theorem MeasureTheory.measure_eq_top_of_setLintegral_ne_top {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {s : Set α} (hf : AEMeasurable f) (hμf : ∫⁻ (x : α) in s, f x ∂μ ≠ ⊤) : ↑↑μ {x | x ∈ s ∧ f x = ⊤} = 0

theorem MeasureTheory.meas_ge_le_lintegral_div {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : AEMeasurable f) {ε : ENNReal} (hε : ε ≠ 0) (hε' : ε ≠ ⊤) : ↑↑μ {x | ε ≤ f x} ≤ (∫⁻ (a : α), f a ∂μ) / ε

Markov's inequality also known as Chebyshev's first inequality.

theorem MeasureTheory.ae_eq_of_ae_le_of_lintegral_le {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (hfg : f ≤ᶠ[MeasureTheory.Measure.ae μ] g) (hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤) (hg : AEMeasurable g) (hgf : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ) : f =ᶠ[MeasureTheory.Measure.ae μ] g

@[simp]

theorem MeasureTheory.lintegral_eq_zero_iff' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : AEMeasurable f) : ∫⁻ (a : α), f a ∂μ = 0 ↔ f =ᶠ[MeasureTheory.Measure.ae μ] 0

@[simp]

theorem MeasureTheory.lintegral_eq_zero_iff {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : Measurable f) : ∫⁻ (a : α), f a ∂μ = 0 ↔ f =ᶠ[MeasureTheory.Measure.ae μ] 0

theorem MeasureTheory.lintegral_pos_iff_support {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : Measurable f) : 0 < ∫⁻ (a : α), f a ∂μ ↔ 0 < ↑↑μ (Function.support f)

theorem MeasureTheory.lintegral_iSup_ae {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ENNReal} (hf : ∀ (n : ℕ), Measurable (f n)) (h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f (Nat.succ n) a) : ∫⁻ (a : α), ⨆ (n : ℕ), f n a ∂μ = ⨆ (n : ℕ), ∫⁻ (a : α), f n a ∂μ

Weaker version of the monotone convergence theorem

theorem MeasureTheory.lintegral_sub' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (hg : AEMeasurable g) (hg_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤) (h_le : g ≤ᶠ[MeasureTheory.Measure.ae μ] f) : ∫⁻ (a : α), f a - g a ∂μ = ∫⁻ (a : α), f a ∂μ - ∫⁻ (a : α), g a ∂μ

theorem MeasureTheory.lintegral_sub {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (hg : Measurable g) (hg_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤) (h_le : g ≤ᶠ[MeasureTheory.Measure.ae μ] f) : ∫⁻ (a : α), f a - g a ∂μ = ∫⁻ (a : α), f a ∂μ - ∫⁻ (a : α), g a ∂μ

theorem MeasureTheory.lintegral_sub_le' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ENNReal) (g : α → ENNReal) (hf : AEMeasurable f) : ∫⁻ (x : α), g x ∂μ - ∫⁻ (x : α), f x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ

theorem MeasureTheory.lintegral_sub_le {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ENNReal) (g : α → ENNReal) (hf : Measurable f) : ∫⁻ (x : α), g x ∂μ - ∫⁻ (x : α), f x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ

theorem MeasureTheory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (hg : AEMeasurable g) (hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤) (h_le : f ≤ᶠ[MeasureTheory.Measure.ae μ] g) (h : ∃ᵐ (x : α) ∂μ, f x ≠ g x) : ∫⁻ (x : α), f x ∂μ < ∫⁻ (x : α), g x ∂μ

theorem MeasureTheory.lintegral_strict_mono_of_ae_le_of_ae_lt_on {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (hg : AEMeasurable g) (hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤) (h_le : f ≤ᶠ[MeasureTheory.Measure.ae μ] g) {s : Set α} (hμs : ↑↑μ s ≠ 0) (h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x < g x) : ∫⁻ (x : α), f x ∂μ < ∫⁻ (x : α), g x ∂μ

theorem MeasureTheory.lintegral_strict_mono {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (hμ : μ ≠ 0) (hg : AEMeasurable g) (hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤) (h : ∀ᵐ (x : α) ∂μ, f x < g x) : ∫⁻ (x : α), f x ∂μ < ∫⁻ (x : α), g x ∂μ

theorem MeasureTheory.set_lintegral_strict_mono {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} {s : Set α} (hsm : MeasurableSet s) (hs : ↑↑μ s ≠ 0) (hg : Measurable g) (hfi : ∫⁻ (x : α) in s, f x ∂μ ≠ ⊤) (h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x < g x) : ∫⁻ (x : α) in s, f x ∂μ < ∫⁻ (x : α) in s, g x ∂μ

theorem MeasureTheory.lintegral_iInf_ae {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ENNReal} (h_meas : ∀ (n : ℕ), Measurable (f n)) (h_mono : ∀ (n : ℕ), f (Nat.succ n) ≤ᶠ[MeasureTheory.Measure.ae μ] f n) (h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤) : ∫⁻ (a : α), ⨅ (n : ℕ), f n a ∂μ = ⨅ (n : ℕ), ∫⁻ (a : α), f n a ∂μ

Monotone convergence theorem for nonincreasing sequences of functions

theorem MeasureTheory.lintegral_iInf {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ENNReal} (h_meas : ∀ (n : ℕ), Measurable (f n)) (h_anti : Antitone f) (h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤) : ∫⁻ (a : α), ⨅ (n : ℕ), f n a ∂μ = ⨅ (n : ℕ), ∫⁻ (a : α), f n a ∂μ

Monotone convergence theorem for nonincreasing sequences of functions

theorem MeasureTheory.lintegral_liminf_le' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ENNReal} (h_meas : ∀ (n : ℕ), AEMeasurable (f n)) : ∫⁻ (a : α), Filter.liminf (fun n => f n a) Filter.atTop ∂μ ≤ Filter.liminf (fun n => ∫⁻ (a : α), f n a ∂μ) Filter.atTop

Known as Fatou's lemma, version with AEMeasurable functions

theorem MeasureTheory.lintegral_liminf_le {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ENNReal} (h_meas : ∀ (n : ℕ), Measurable (f n)) : ∫⁻ (a : α), Filter.liminf (fun n => f n a) Filter.atTop ∂μ ≤ Filter.liminf (fun n => ∫⁻ (a : α), f n a ∂μ) Filter.atTop

Known as Fatou's lemma

theorem MeasureTheory.limsup_lintegral_le {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ENNReal} {g : α → ENNReal} (hf_meas : ∀ (n : ℕ), Measurable (f n)) (h_bound : ∀ (n : ℕ), f n ≤ᶠ[MeasureTheory.Measure.ae μ] g) (h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤) : Filter.limsup (fun n => ∫⁻ (a : α), f n a ∂μ) Filter.atTop ≤ ∫⁻ (a : α), Filter.limsup (fun n => f n a) Filter.atTop ∂μ

theorem MeasureTheory.tendsto_lintegral_of_dominated_convergence {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {F : ℕ → α → ENNReal} {f : α → ENNReal} (bound : α → ENNReal) (hF_meas : ∀ (n : ℕ), Measurable (F n)) (h_bound : ∀ (n : ℕ), F n ≤ᶠ[MeasureTheory.Measure.ae μ] bound) (h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) Filter.atTop (nhds (f a))) : Filter.Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) Filter.atTop (nhds (∫⁻ (a : α), f a ∂μ))

Dominated convergence theorem for nonnegative functions

theorem MeasureTheory.tendsto_lintegral_of_dominated_convergence' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {F : ℕ → α → ENNReal} {f : α → ENNReal} (bound : α → ENNReal) (hF_meas : ∀ (n : ℕ), AEMeasurable (F n)) (h_bound : ∀ (n : ℕ), F n ≤ᶠ[MeasureTheory.Measure.ae μ] bound) (h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) Filter.atTop (nhds (f a))) : Filter.Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) Filter.atTop (nhds (∫⁻ (a : α), f a ∂μ))

Dominated convergence theorem for nonnegative functions which are just almost everywhere measurable.

theorem MeasureTheory.tendsto_lintegral_filter_of_dominated_convergence {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_5} {l : Filter ι} [Filter.IsCountablyGenerated l] {F : ι → α → ENNReal} {f : α → ENNReal} (bound : α → ENNReal) (hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n)) (h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) (h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) l (nhds (f a))) : Filter.Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) l (nhds (∫⁻ (a : α), f a ∂μ))

Dominated convergence theorem for filters with a countable basis

theorem MeasureTheory.lintegral_iSup_directed_of_measurable {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable β] {f : β → α → ENNReal} (hf : ∀ (b : β), Measurable (f b)) (h_directed : Directed (fun x x_1 => x ≤ x_1) f) : ∫⁻ (a : α), ⨆ (b : β), f b a ∂μ = ⨆ (b : β), ∫⁻ (a : α), f b a ∂μ

Monotone convergence for a supremum over a directed family and indexed by a countable type

theorem MeasureTheory.lintegral_iSup_directed {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable β] {f : β → α → ENNReal} (hf : ∀ (b : β), AEMeasurable (f b)) (h_directed : Directed (fun x x_1 => x ≤ x_1) f) : ∫⁻ (a : α), ⨆ (b : β), f b a ∂μ = ⨆ (b : β), ∫⁻ (a : α), f b a ∂μ

Monotone convergence for a supremum over a directed family and indexed by a countable type.

theorem MeasureTheory.lintegral_tsum {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable β] {f : β → α → ENNReal} (hf : ∀ (i : β), AEMeasurable (f i)) : ∫⁻ (a : α), ∑' (i : β), f i a ∂μ = ∑' (i : β), ∫⁻ (a : α), f i a ∂μ

theorem MeasureTheory.lintegral_iUnion₀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable β] {s : β → Set α} (hm : ∀ (i : β), MeasureTheory.NullMeasurableSet (s i)) (hd : Pairwise (MeasureTheory.AEDisjoint μ on s)) (f : α → ENNReal) : ∫⁻ (a : α) in ⋃ (i : β), s i, f a ∂μ = ∑' (i : β), ∫⁻ (a : α) in s i, f a ∂μ

theorem MeasureTheory.lintegral_iUnion {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable β] {s : β → Set α} (hm : ∀ (i : β), MeasurableSet (s i)) (hd : Pairwise (Disjoint on s)) (f : α → ENNReal) : ∫⁻ (a : α) in ⋃ (i : β), s i, f a ∂μ = ∑' (i : β), ∫⁻ (a : α) in s i, f a ∂μ

theorem MeasureTheory.lintegral_biUnion₀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set β} {s : β → Set α} (ht : Set.Countable t) (hm : ∀ (i : β), i ∈ t → MeasureTheory.NullMeasurableSet (s i)) (hd : Set.Pairwise t (MeasureTheory.AEDisjoint μ on s)) (f : α → ENNReal) : ∫⁻ (a : α) in ⋃ (i : β) (_ : i ∈ t), s i, f a ∂μ = ∑' (i : ↑t), ∫⁻ (a : α) in s ↑i, f a ∂μ

theorem MeasureTheory.lintegral_biUnion {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set β} {s : β → Set α} (ht : Set.Countable t) (hm : ∀ (i : β), i ∈ t → MeasurableSet (s i)) (hd : Set.PairwiseDisjoint t s) (f : α → ENNReal) : ∫⁻ (a : α) in ⋃ (i : β) (_ : i ∈ t), s i, f a ∂μ = ∑' (i : ↑t), ∫⁻ (a : α) in s ↑i, f a ∂μ

theorem MeasureTheory.lintegral_biUnion_finset₀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Finset β} {t : β → Set α} (hd : Set.Pairwise (↑s) (MeasureTheory.AEDisjoint μ on t)) (hm : ∀ (b : β), b ∈ s → MeasureTheory.NullMeasurableSet (t b)) (f : α → ENNReal) : ∫⁻ (a : α) in ⋃ (b : β) (_ : b ∈ s), t b, f a ∂μ = Finset.sum s fun b => ∫⁻ (a : α) in t b, f a ∂μ

theorem MeasureTheory.lintegral_biUnion_finset {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Finset β} {t : β → Set α} (hd : Set.PairwiseDisjoint (↑s) t) (hm : ∀ (b : β), b ∈ s → MeasurableSet (t b)) (f : α → ENNReal) : ∫⁻ (a : α) in ⋃ (b : β) (_ : b ∈ s), t b, f a ∂μ = Finset.sum s fun b => ∫⁻ (a : α) in t b, f a ∂μ

theorem MeasureTheory.lintegral_iUnion_le {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable β] (s : β → Set α) (f : α → ENNReal) : ∫⁻ (a : α) in ⋃ (i : β), s i, f a ∂μ ≤ ∑' (i : β), ∫⁻ (a : α) in s i, f a ∂μ

theorem MeasureTheory.lintegral_union {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {A : Set α} {B : Set α} (hB : MeasurableSet B) (hAB : Disjoint A B) : ∫⁻ (a : α) in A ∪ B, f a ∂μ = ∫⁻ (a : α) in A, f a ∂μ + ∫⁻ (a : α) in B, f a ∂μ

theorem MeasureTheory.lintegral_union_le {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ENNReal) (s : Set α) (t : Set α) : ∫⁻ (a : α) in s ∪ t, f a ∂μ ≤ ∫⁻ (a : α) in s, f a ∂μ + ∫⁻ (a : α) in t, f a ∂μ

theorem MeasureTheory.lintegral_inter_add_diff {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {B : Set α} (f : α → ENNReal) (A : Set α) (hB : MeasurableSet B) : ∫⁻ (x : α) in A ∩ B, f x ∂μ + ∫⁻ (x : α) in A \ B, f x ∂μ = ∫⁻ (x : α) in A, f x ∂μ

theorem MeasureTheory.lintegral_add_compl {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ENNReal) {A : Set α} (hA : MeasurableSet A) : ∫⁻ (x : α) in A, f x ∂μ + ∫⁻ (x : α) in Aᶜ, f x ∂μ = ∫⁻ (x : α), f x ∂μ

theorem MeasureTheory.lintegral_max {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (hf : Measurable f) (hg : Measurable g) : ∫⁻ (x : α), max (f x) (g x) ∂μ = ∫⁻ (x : α) in {x | f x ≤ g x}, g x ∂μ + ∫⁻ (x : α) in {x | g x < f x}, f x ∂μ

theorem MeasureTheory.set_lintegral_max {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (hf : Measurable f) (hg : Measurable g) (s : Set α) : ∫⁻ (x : α) in s, max (f x) (g x) ∂μ = ∫⁻ (x : α) in s ∩ {x | f x ≤ g x}, g x ∂μ + ∫⁻ (x : α) in s ∩ {x | g x < f x}, f x ∂μ

theorem MeasureTheory.lintegral_map {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mβ : MeasurableSpace β} {f : β → ENNReal} {g : α → β} (hf : Measurable f) (hg : Measurable g) : ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ = ∫⁻ (a : α), f (g a) ∂μ

theorem MeasureTheory.lintegral_map' {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mβ : MeasurableSpace β} {f : β → ENNReal} {g : α → β} (hf : AEMeasurable f) (hg : AEMeasurable g) : ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ = ∫⁻ (a : α), f (g a) ∂μ

theorem MeasureTheory.lintegral_map_le {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mβ : MeasurableSpace β} (f : β → ENNReal) {g : α → β} (hg : Measurable g) : ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ ≤ ∫⁻ (a : α), f (g a) ∂μ

theorem MeasureTheory.lintegral_comp {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace β] {f : β → ENNReal} {g : α → β} (hf : Measurable f) (hg : Measurable g) : MeasureTheory.lintegral μ (f ∘ g) = ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ

theorem MeasureTheory.set_lintegral_map {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace β] {f : β → ENNReal} {g : α → β} {s : Set β} (hs : MeasurableSet s) (hf : Measurable f) (hg : Measurable g) : ∫⁻ (y : β) in s, f y ∂MeasureTheory.Measure.map g μ = ∫⁻ (x : α) in g ⁻¹' s, f (g x) ∂μ

theorem MeasureTheory.lintegral_indicator_const_comp {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mβ : MeasurableSpace β} {f : α → β} {s : Set β} (hf : Measurable f) (hs : MeasurableSet s) (c : ENNReal) : ∫⁻ (a : α), Set.indicator s (fun x => c) (f a) ∂μ = c * ↑↑μ (f ⁻¹' s)

theorem MeasurableEmbedding.lintegral_map {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace β] {g : α → β} (hg : MeasurableEmbedding g) (f : β → ENNReal) : ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ = ∫⁻ (a : α), f (g a) ∂μ

If g : α → β is a measurable embedding and f : β → ℝ≥0∞ is any function (not necessarily measurable), then ∫⁻ a, f a ∂(map g μ) = ∫⁻ a, f (g a) ∂μ. Compare with lintegral_map which applies to any measurable g : α → β but requires that f is measurable as well.

theorem MeasureTheory.lintegral_map_equiv {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace β] (f : β → ENNReal) (g : α ≃ᵐ β) : ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map (↑g) μ = ∫⁻ (a : α), f (↑g a) ∂μ

The lintegral transforms appropriately under a measurable equivalence g : α ≃ᵐ β. (Compare lintegral_map, which applies to a wider class of functions g : α → β, but requires measurability of the function being integrated.)

theorem MeasureTheory.MeasurePreserving.lintegral_comp {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mb : MeasurableSpace β} {ν : MeasureTheory.Measure β} {g : α → β} (hg : MeasureTheory.MeasurePreserving g) {f : β → ENNReal} (hf : Measurable f) : ∫⁻ (a : α), f (g a) ∂μ = ∫⁻ (b : β), f b ∂ν

theorem MeasureTheory.MeasurePreserving.lintegral_comp_emb {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mb : MeasurableSpace β} {ν : MeasureTheory.Measure β} {g : α → β} (hg : MeasureTheory.MeasurePreserving g) (hge : MeasurableEmbedding g) (f : β → ENNReal) : ∫⁻ (a : α), f (g a) ∂μ = ∫⁻ (b : β), f b ∂ν

theorem MeasureTheory.MeasurePreserving.set_lintegral_comp_preimage {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mb : MeasurableSpace β} {ν : MeasureTheory.Measure β} {g : α → β} (hg : MeasureTheory.MeasurePreserving g) {s : Set β} (hs : MeasurableSet s) {f : β → ENNReal} (hf : Measurable f) : ∫⁻ (a : α) in g ⁻¹' s, f (g a) ∂μ = ∫⁻ (b : β) in s, f b ∂ν

theorem MeasureTheory.MeasurePreserving.set_lintegral_comp_preimage_emb {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mb : MeasurableSpace β} {ν : MeasureTheory.Measure β} {g : α → β} (hg : MeasureTheory.MeasurePreserving g) (hge : MeasurableEmbedding g) (f : β → ENNReal) (s : Set β) : ∫⁻ (a : α) in g ⁻¹' s, f (g a) ∂μ = ∫⁻ (b : β) in s, f b ∂ν

theorem MeasureTheory.MeasurePreserving.set_lintegral_comp_emb {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mb : MeasurableSpace β} {ν : MeasureTheory.Measure β} {g : α → β} (hg : MeasureTheory.MeasurePreserving g) (hge : MeasurableEmbedding g) (f : β → ENNReal) (s : Set α) : ∫⁻ (a : α) in s, f (g a) ∂μ = ∫⁻ (b : β) in g '' s, f b ∂ν

instance MeasurableSpace.Top.measurableSingletonClass {α : Type u_5} : MeasurableSingletonClass α

theorem MeasureTheory.lintegral_dirac' {α : Type u_1} [MeasurableSpace α] (a : α) {f : α → ENNReal} (hf : Measurable f) : ∫⁻ (a : α), f a ∂MeasureTheory.Measure.dirac a = f a

theorem MeasureTheory.lintegral_dirac {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (a : α) (f : α → ENNReal) : ∫⁻ (a : α), f a ∂MeasureTheory.Measure.dirac a = f a

theorem MeasureTheory.set_lintegral_dirac' {α : Type u_1} [MeasurableSpace α] {a : α} {f : α → ENNReal} (hf : Measurable f) {s : Set α} (hs : MeasurableSet s) [Decidable (a ∈ s)] : ∫⁻ (x : α) in s, f x ∂MeasureTheory.Measure.dirac a = if a ∈ s then f a else 0

theorem MeasureTheory.set_lintegral_dirac {α : Type u_1} [MeasurableSpace α] {a : α} (f : α → ENNReal) (s : Set α) [MeasurableSingletonClass α] [Decidable (a ∈ s)] : ∫⁻ (x : α) in s, f x ∂MeasureTheory.Measure.dirac a = if a ∈ s then f a else 0

theorem MeasureTheory.lintegral_count' {α : Type u_1} [MeasurableSpace α] {f : α → ENNReal} (hf : Measurable f) : ∫⁻ (a : α), f a ∂MeasureTheory.Measure.count = ∑' (a : α), f a

theorem MeasureTheory.lintegral_count {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (f : α → ENNReal) : ∫⁻ (a : α), f a ∂MeasureTheory.Measure.count = ∑' (a : α), f a

theorem ENNReal.tsum_const_eq {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (c : ENNReal) : ∑' (x : α), c = c * ↑↑MeasureTheory.Measure.count Set.univ

theorem ENNReal.count_const_le_le_of_tsum_le {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] {a : α → ENNReal} (a_mble : Measurable a) {c : ENNReal} (tsum_le_c : ∑' (i : α), a i ≤ c) {ε : ENNReal} (ε_ne_zero : ε ≠ 0) (ε_ne_top : ε ≠ ⊤) : ↑↑MeasureTheory.Measure.count {i | ε ≤ a i} ≤ c / ε

Markov's inequality for the counting measure with hypothesis using tsum in ℝ≥0∞.

theorem NNReal.count_const_le_le_of_tsum_le {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] {a : α → NNReal} (a_mble : Measurable a) (a_summable : Summable a) {c : NNReal} (tsum_le_c : ∑' (i : α), a i ≤ c) {ε : NNReal} (ε_ne_zero : ε ≠ 0) : ↑↑MeasureTheory.Measure.count {i | ε ≤ a i} ≤ ↑c / ↑ε

Markov's inequality for counting measure with hypothesis using tsum in ℝ≥0.

Lebesgue integral over finite and countable types and sets

theorem MeasureTheory.lintegral_countable' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable α] [MeasurableSingletonClass α] (f : α → ENNReal) : ∫⁻ (a : α), f a ∂μ = ∑' (a : α), f a * ↑↑μ {a}

theorem MeasureTheory.lintegral_singleton' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : Measurable f) (a : α) : ∫⁻ (x : α) in {a}, f x ∂μ = f a * ↑↑μ {a}

theorem MeasureTheory.lintegral_singleton {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass α] (f : α → ENNReal) (a : α) : ∫⁻ (x : α) in {a}, f x ∂μ = f a * ↑↑μ {a}

theorem MeasureTheory.lintegral_countable {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass α] (f : α → ENNReal) {s : Set α} (hs : Set.Countable s) : ∫⁻ (a : α) in s, f a ∂μ = ∑' (a : ↑s), f ↑a * ↑↑μ {↑a}

theorem MeasureTheory.lintegral_insert {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass α] {a : α} {s : Set α} (h : ¬a ∈ s) (f : α → ENNReal) : ∫⁻ (x : α) in insert a s, f x ∂μ = f a * ↑↑μ {a} + ∫⁻ (x : α) in s, f x ∂μ

theorem MeasureTheory.lintegral_finset {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass α] (s : Finset α) (f : α → ENNReal) : ∫⁻ (x : α) in ↑s, f x ∂μ = Finset.sum s fun x => f x * ↑↑μ {x}

theorem MeasureTheory.lintegral_fintype {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass α] [Fintype α] (f : α → ENNReal) : ∫⁻ (x : α), f x ∂μ = Finset.sum Finset.univ fun x => f x * ↑↑μ {x}

theorem MeasureTheory.lintegral_unique {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Unique α] (f : α → ENNReal) : ∫⁻ (x : α), f x ∂μ = f default * ↑↑μ Set.univ

theorem MeasureTheory.ae_lt_top {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : Measurable f) (h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤) : ∀ᵐ (x : α) ∂μ, f x < ⊤

theorem MeasureTheory.ae_lt_top' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : AEMeasurable f) (h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤) : ∀ᵐ (x : α) ∂μ, f x < ⊤

theorem MeasureTheory.set_lintegral_lt_top_of_bddAbove {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : ↑↑μ s ≠ ⊤) {f : α → NNReal} (hf : Measurable f) (hbdd : BddAbove (f '' s)) : ∫⁻ (x : α) in s, ↑(f x) ∂μ < ⊤

theorem MeasureTheory.set_lintegral_lt_top_of_isCompact {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [OpensMeasurableSpace α] {s : Set α} (hs : ↑↑μ s ≠ ⊤) (hsc : IsCompact s) {f : α → NNReal} (hf : Continuous f) : ∫⁻ (x : α) in s, ↑(f x) ∂μ < ⊤

theorem IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal {α : Type u_5} [MeasurableSpace α] (μ : MeasureTheory.Measure α) [μ_fin : MeasureTheory.IsFiniteMeasure μ] {f : α → ENNReal} (f_bdd : ∃ c, ∀ (x : α), f x ≤ ↑c) : ∫⁻ (x : α), f x ∂μ < ⊤

def MeasureTheory.Measure.withDensity {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ENNReal) : MeasureTheory.Measure α

Given a measure μ : Measure α and a function f : α → ℝ≥0∞, μ.withDensity f is the measure such that for a measurable set s we have μ.withDensity f s = ∫⁻ a in s, f a ∂μ.

@[simp]

theorem MeasureTheory.withDensity_apply {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ENNReal) {s : Set α} (hs : MeasurableSet s) : ↑↑(MeasureTheory.Measure.withDensity μ f) s = ∫⁻ (a : α) in s, f a ∂μ

theorem MeasureTheory.withDensity_congr_ae {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (h : f =ᶠ[MeasureTheory.Measure.ae μ] g) : MeasureTheory.Measure.withDensity μ f = MeasureTheory.Measure.withDensity μ g

theorem MeasureTheory.withDensity_add_left {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : Measurable f) (g : α → ENNReal) : MeasureTheory.Measure.withDensity μ (f + g) = MeasureTheory.Measure.withDensity μ f + MeasureTheory.Measure.withDensity μ g

theorem MeasureTheory.withDensity_add_right {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ENNReal) {g : α → ENNReal} (hg : Measurable g) : MeasureTheory.Measure.withDensity μ (f + g) = MeasureTheory.Measure.withDensity μ f + MeasureTheory.Measure.withDensity μ g

theorem MeasureTheory.withDensity_add_measure {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) (f : α → ENNReal) : MeasureTheory.Measure.withDensity (μ + ν) f = MeasureTheory.Measure.withDensity μ f + MeasureTheory.Measure.withDensity ν f

theorem MeasureTheory.withDensity_sum {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} (μ : ι → MeasureTheory.Measure α) (f : α → ENNReal) : MeasureTheory.Measure.withDensity (MeasureTheory.Measure.sum μ) f = MeasureTheory.Measure.sum fun n => MeasureTheory.Measure.withDensity (μ n) f

theorem MeasureTheory.withDensity_smul {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ENNReal) {f : α → ENNReal} (hf : Measurable f) : MeasureTheory.Measure.withDensity μ (r • f) = r • MeasureTheory.Measure.withDensity μ f

theorem MeasureTheory.withDensity_smul' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ENNReal) (f : α → ENNReal) (hr : r ≠ ⊤) : MeasureTheory.Measure.withDensity μ (r • f) = r • MeasureTheory.Measure.withDensity μ f

theorem MeasureTheory.isFiniteMeasure_withDensity {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : ∫⁻ (a : α), f a ∂μ ≠ ⊤) : MeasureTheory.IsFiniteMeasure (MeasureTheory.Measure.withDensity μ f)

theorem MeasureTheory.withDensity_absolutelyContinuous {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ENNReal) : MeasureTheory.Measure.AbsolutelyContinuous (MeasureTheory.Measure.withDensity μ f) μ

@[simp]

theorem MeasureTheory.withDensity_zero {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} : MeasureTheory.Measure.withDensity μ 0 = 0

@[simp]

theorem MeasureTheory.withDensity_one {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} : MeasureTheory.Measure.withDensity μ 1 = μ

theorem MeasureTheory.withDensity_tsum {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ENNReal} (h : ∀ (i : ℕ), Measurable (f i)) : MeasureTheory.Measure.withDensity μ (∑' (n : ℕ), f n) = MeasureTheory.Measure.sum fun n => MeasureTheory.Measure.withDensity μ (f n)

theorem MeasureTheory.withDensity_indicator {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (f : α → ENNReal) : MeasureTheory.Measure.withDensity μ (Set.indicator s f) = MeasureTheory.Measure.withDensity (MeasureTheory.Measure.restrict μ s) f

theorem MeasureTheory.withDensity_indicator_one {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) : MeasureTheory.Measure.withDensity μ (Set.indicator s 1) = MeasureTheory.Measure.restrict μ s

theorem MeasureTheory.withDensity_ofReal_mutuallySingular {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : Measurable f) : MeasureTheory.Measure.MutuallySingular (MeasureTheory.Measure.withDensity μ fun x => ENNReal.ofReal (f x)) (MeasureTheory.Measure.withDensity μ fun x => ENNReal.ofReal (-f x))

theorem MeasureTheory.restrict_withDensity {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (f : α → ENNReal) : MeasureTheory.Measure.restrict (MeasureTheory.Measure.withDensity μ f) s = MeasureTheory.Measure.withDensity (MeasureTheory.Measure.restrict μ s) f

theorem MeasureTheory.withDensity_eq_zero {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : AEMeasurable f) (h : MeasureTheory.Measure.withDensity μ f = 0) : f =ᶠ[MeasureTheory.Measure.ae μ] 0

theorem MeasureTheory.withDensity_apply_eq_zero {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {s : Set α} (hf : Measurable f) : ↑↑(MeasureTheory.Measure.withDensity μ f) s = 0 ↔ ↑↑μ ({x | f x ≠ 0} ∩ s) = 0

theorem MeasureTheory.ae_withDensity_iff {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : α → Prop} {f : α → ENNReal} (hf : Measurable f) : (∀ᵐ (x : α) ∂MeasureTheory.Measure.withDensity μ f, p x) ↔ ∀ᵐ (x : α) ∂μ, f x ≠ 0 → p x

theorem MeasureTheory.ae_withDensity_iff_ae_restrict {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : α → Prop} {f : α → ENNReal} (hf : Measurable f) : (∀ᵐ (x : α) ∂MeasureTheory.Measure.withDensity μ f, p x) ↔ ∀ᵐ (x : α) ∂MeasureTheory.Measure.restrict μ {x | f x ≠ 0}, p x

theorem MeasureTheory.aemeasurable_withDensity_ennreal_iff {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → NNReal} (hf : Measurable f) {g : α → ENNReal} : AEMeasurable g ↔ AEMeasurable fun x => ↑(f x) * g x

theorem MeasureTheory.lintegral_withDensity_eq_lintegral_mul {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ENNReal} (h_mf : Measurable f) {g : α → ENNReal} : Measurable g → ∫⁻ (a : α), g a ∂MeasureTheory.Measure.withDensity μ f = ∫⁻ (a : α), (f * g) a ∂μ

This is Exercise 1.2.1 from [tao2010]. It allows you to express integration of a measurable function with respect to (μ.withDensity f) as an integral with respect to μ, called the base measure. μ is often the Lebesgue measure, and in this circumstance f is the probability density function, and (μ.withDensity f) represents any continuous random variable as a probability measure, such as the uniform distribution between 0 and 1, the Gaussian distribution, the exponential distribution, the Beta distribution, or the Cauchy distribution (see Section 2.4 of [wasserman2004]). Thus, this method shows how to one can calculate expectations, variances, and other moments as a function of the probability density function.

theorem MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ENNReal} {g : α → ENNReal} (hf : Measurable f) (hg : Measurable g) {s : Set α} (hs : MeasurableSet s) : ∫⁻ (x : α) in s, g x ∂MeasureTheory.Measure.withDensity μ f = ∫⁻ (x : α) in s, (f * g) x ∂μ

theorem MeasureTheory.lintegral_withDensity_eq_lintegral_mul₀' {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : AEMeasurable f) {g : α → ENNReal} (hg : AEMeasurable g) : ∫⁻ (a : α), g a ∂MeasureTheory.Measure.withDensity μ f = ∫⁻ (a : α), (f * g) a ∂μ

The Lebesgue integral of g with respect to the measure μ.withDensity f coincides with the integral of f * g. This version assumes that g is almost everywhere measurable. For a version without conditions on g but requiring that f is almost everywhere finite, see lintegral_withDensity_eq_lintegral_mul_non_measurable

theorem MeasureTheory.lintegral_withDensity_eq_lintegral_mul₀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : AEMeasurable f) {g : α → ENNReal} (hg : AEMeasurable g) : ∫⁻ (a : α), g a ∂MeasureTheory.Measure.withDensity μ f = ∫⁻ (a : α), (f * g) a ∂μ

theorem MeasureTheory.lintegral_withDensity_le_lintegral_mul {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ENNReal} (f_meas : Measurable f) (g : α → ENNReal) : ∫⁻ (a : α), g a ∂MeasureTheory.Measure.withDensity μ f ≤ ∫⁻ (a : α), (f * g) a ∂μ

theorem MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ENNReal} (f_meas : Measurable f) (hf : ∀ᵐ (x : α) ∂μ, f x < ⊤) (g : α → ENNReal) : ∫⁻ (a : α), g a ∂MeasureTheory.Measure.withDensity μ f = ∫⁻ (a : α), (f * g) a ∂μ

theorem MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ENNReal} (f_meas : Measurable f) (g : α → ENNReal) {s : Set α} (hs : MeasurableSet s) (hf : ∀ᵐ (x : α) ∂MeasureTheory.Measure.restrict μ s, f x < ⊤) : ∫⁻ (a : α) in s, g a ∂MeasureTheory.Measure.withDensity μ f = ∫⁻ (a : α) in s, (f * g) a ∂μ

theorem MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable₀ {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ENNReal} (hf : AEMeasurable f) (h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤) (g : α → ENNReal) : ∫⁻ (a : α), g a ∂MeasureTheory.Measure.withDensity μ f = ∫⁻ (a : α), (f * g) a ∂μ

theorem MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable₀ {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ENNReal} {s : Set α} (hf : AEMeasurable f) (g : α → ENNReal) (hs : MeasurableSet s) (h'f : ∀ᵐ (x : α) ∂MeasureTheory.Measure.restrict μ s, f x < ⊤) : ∫⁻ (a : α) in s, g a ∂MeasureTheory.Measure.withDensity μ f = ∫⁻ (a : α) in s, (f * g) a ∂μ

theorem MeasureTheory.withDensity_mul {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ENNReal} {g : α → ENNReal} (hf : Measurable f) (hg : Measurable g) : MeasureTheory.Measure.withDensity μ (f * g) = MeasureTheory.Measure.withDensity (MeasureTheory.Measure.withDensity μ f) g

theorem MeasureTheory.exists_pos_lintegral_lt_of_sigmaFinite {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] {ε : ENNReal} (ε0 : ε ≠ 0) : ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε

In a sigma-finite measure space, there exists an integrable function which is positive everywhere (and with an arbitrarily small integral).

theorem MeasureTheory.lintegral_trim {α : Type u_1} {m : MeasurableSpace α} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) {f : α → ENNReal} (hf : Measurable f) : ∫⁻ (a : α), f a ∂MeasureTheory.Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ

theorem MeasureTheory.lintegral_trim_ae {α : Type u_1} {m : MeasurableSpace α} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) {f : α → ENNReal} (hf : AEMeasurable f) : ∫⁻ (a : α), f a ∂MeasureTheory.Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ

theorem MeasureTheory.univ_le_of_forall_fin_meas_le {α : Type u_1} {m : MeasurableSpace α} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) [MeasureTheory.SigmaFinite (MeasureTheory.Measure.trim μ hm)] (C : ENNReal) {f : Set α → ENNReal} (hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → f s ≤ C) (h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ (n : ℕ), S n) ≤ ⨆ (n : ℕ), f (S n)) : f Set.univ ≤ C

theorem MeasureTheory.lintegral_le_of_forall_fin_meas_le_of_measurable {α : Type u_1} {m : MeasurableSpace α} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) [MeasureTheory.SigmaFinite (MeasureTheory.Measure.trim μ hm)] (C : ENNReal) {f : α → ENNReal} (hf_meas : Measurable f) (hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C) : ∫⁻ (x : α), f x ∂μ ≤ C

If the Lebesgue integral of a function is bounded by some constant on all sets with finite measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral over the whole space is bounded by that same constant. Version for a measurable function. See lintegral_le_of_forall_fin_meas_le' for the more general AEMeasurable version.

theorem MeasureTheory.lintegral_le_of_forall_fin_meas_le' {α : Type u_1} {m : MeasurableSpace α} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) [MeasureTheory.SigmaFinite (MeasureTheory.Measure.trim μ hm)] (C : ENNReal) {f : α → ENNReal} (hf_meas : AEMeasurable f) (hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C) : ∫⁻ (x : α), f x ∂μ ≤ C

If the Lebesgue integral of a function is bounded by some constant on all sets with finite measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral over the whole space is bounded by that same constant.

theorem MeasureTheory.lintegral_le_of_forall_fin_meas_le {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] (C : ENNReal) {f : α → ENNReal} (hf_meas : AEMeasurable f) (hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C) : ∫⁻ (x : α), f x ∂μ ≤ C

If the Lebesgue integral of a function is bounded by some constant on all sets with finite measure and the measure is σ-finite, then the integral over the whole space is bounded by that same constant.

theorem MeasureTheory.SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f : MeasureTheory.SimpleFunc α NNReal} {L : ENNReal} (hL : L < ∫⁻ (x : α), ↑(↑f x) ∂μ) : ∃ g, (∀ (x : α), ↑g x ≤ ↑f x) ∧ ∫⁻ (x : α), ↑(↑g x) ∂μ < ⊤ ∧ L < ∫⁻ (x : α), ↑(↑g x) ∂μ

theorem MeasureTheory.exists_lt_lintegral_simpleFunc_of_lt_lintegral {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f : α → NNReal} {L : ENNReal} (hL : L < ∫⁻ (x : α), ↑(f x) ∂μ) : ∃ g, (∀ (x : α), ↑g x ≤ f x) ∧ ∫⁻ (x : α), ↑(↑g x) ∂μ < ⊤ ∧ L < ∫⁻ (x : α), ↑(↑g x) ∂μ

theorem MeasureTheory.exists_absolutelyContinuous_isFiniteMeasure {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] : ∃ ν, MeasureTheory.IsFiniteMeasure ν ∧ MeasureTheory.Measure.AbsolutelyContinuous μ ν

A sigma-finite measure is absolutely continuous with respect to some finite measure.

theorem MeasureTheory.tendsto_measure_of_ae_tendsto_indicator {α : Type u_5} [MeasurableSpace α] {A : Set α} {ι : Type u_6} (L : Filter ι) [Filter.IsCountablyGenerated L] {As : ι → Set α} {μ : MeasureTheory.Measure α} (A_mble : MeasurableSet A) (As_mble : ∀ (i : ι), MeasurableSet (As i)) {B : Set α} (B_mble : MeasurableSet B) (B_finmeas : ↑↑μ B ≠ ⊤) (As_le_B : ∀ᶠ (i : ι) in L, As i ⊆ B) (h_lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun i => Set.indicator (As i) 1 x) L (nhds (Set.indicator A 1 x))) : Filter.Tendsto (fun i => ↑↑μ (As i)) L (nhds (↑↑μ A))

If the indicators of measurable sets Aᵢ tend pointwise almost everywhere to the indicator of a measurable set A and we eventually have Aᵢ ⊆ B for some set B of finite measure, then the measures of Aᵢ tend to the measure of A.

theorem MeasureTheory.tendsto_measure_of_ae_tendsto_indicator_of_isFiniteMeasure {α : Type u_5} [MeasurableSpace α] {A : Set α} {ι : Type u_6} (L : Filter ι) {As : ι → Set α} [Filter.IsCountablyGenerated L] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] (A_mble : MeasurableSet A) (As_mble : ∀ (i : ι), MeasurableSet (As i)) (h_lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun i => Set.indicator (As i) 1 x) L (nhds (Set.indicator A 1 x))) : Filter.Tendsto (fun i => ↑↑μ (As i)) L (nhds (↑↑μ A))

If μ is a finite measure and the indicators of measurable sets Aᵢ tend pointwise almost everywhere to the indicator of a measurable set A, then the measures μ Aᵢ tend to the measure μ A.