Integral average of a function

In this file we define MeasureTheory.average μ f (notation: ⨍ x, f x ∂μ) to be the average value of f with respect to measure μ. It is defined as ∫ x, f x ∂((μ univ)⁻¹ • μ), so it is equal to zero if f is not integrable or if μ is an infinite measure. If μ is a probability measure, then the average of any function is equal to its integral.

For the average on a set, we use ⨍ x in s, f x ∂μ (notation for ⨍ x, f x ∂(μ.restrict s)). For average w.r.t. the volume, one can omit ∂volume.

Both have a version for the Lebesgue integral rather than Bochner.

We prove several version of the first moment method: An integrable function is below/above its average on a set of positive measure.

Implementation notes

The average is defined as an integral over (μ univ)⁻¹ • μ so that all theorems about Bochner integrals work for the average without modifications. For theorems that require integrability of a function, we provide a convenience lemma MeasureTheory.Integrable.to_average.

TODO

Provide the first moment method for the Lebesgue integral as well. A draft is available on branch first_moment_lintegral in mathlib3 repository.

Tags

integral, center mass, average value

Average value of a function w.r.t. a measure

The (Bochner, Lebesgue) average value of a function f w.r.t. a measure μ (notation: ⨍ x, f x ∂μ, ⨍⁻ x, f x ∂μ) is defined as the (Bochner, Lebesgue) integral divided by the total measure, so it is equal to zero if μ is an infinite measure, and (typically) equal to infinity if f is not integrable. If μ is a probability measure, then the average of any function is equal to its integral.

noncomputable def MeasureTheory.laverage {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ENNReal) : ENNReal

Average value of an ℝ≥0∞-valued function f w.r.t. a measure μ, notation: ⨍⁻ x, f x ∂μ. It is defined as μ univ⁻¹ * ∫⁻ x, f x ∂μ, so it is equal to zero if μ is an infinite measure. If μ is a probability measure, then the average of any function is equal to its integral.

For the average on a set, use ⨍⁻ x in s, f x ∂μ (defined as ⨍⁻ x, f x ∂(μ.restrict s)). For average w.r.t. the volume, one can omit ∂volume.

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

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

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

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

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

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

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

@[simp]

theorem MeasureTheory.laverage_zero {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) : ⨍⁻ (_x : α), 0 ∂μ = 0

@[simp]

theorem MeasureTheory.laverage_zero_measure {α : Type u_1} {m0 : MeasurableSpace α} (f : α → ENNReal) : ⨍⁻ (x : α), f x ∂0 = 0

theorem MeasureTheory.laverage_eq' {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ENNReal) : ⨍⁻ (x : α), f x ∂μ = ∫⁻ (x : α), f x ∂(↑↑μ Set.univ)⁻¹ • μ

theorem MeasureTheory.laverage_eq {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ENNReal) : ⨍⁻ (x : α), f x ∂μ = (∫⁻ (x : α), f x ∂μ) / ↑↑μ Set.univ

theorem MeasureTheory.laverage_eq_lintegral {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] (f : α → ENNReal) : ⨍⁻ (x : α), f x ∂μ = ∫⁻ (x : α), f x ∂μ

@[simp]

theorem MeasureTheory.measure_mul_laverage {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → ENNReal) : ↑↑μ Set.univ * ⨍⁻ (x : α), f x ∂μ = ∫⁻ (x : α), f x ∂μ

theorem MeasureTheory.setLaverage_eq {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ENNReal) (s : Set α) : ⨍⁻ (x : α) in s, f x ∂μ = (∫⁻ (x : α) in s, f x ∂μ) / ↑↑μ s

theorem MeasureTheory.setLaverage_eq' {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ENNReal) (s : Set α) : ⨍⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α), f x ∂(↑↑μ s)⁻¹ • MeasureTheory.Measure.restrict μ s

theorem MeasureTheory.laverage_congr {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (h : f =ᶠ[MeasureTheory.Measure.ae μ] g) : ⨍⁻ (x : α), f x ∂μ = ⨍⁻ (x : α), g x ∂μ

theorem MeasureTheory.setLaverage_congr {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {t : Set α} {f : α → ENNReal} (h : s =ᶠ[MeasureTheory.Measure.ae μ] t) : ⨍⁻ (x : α) in s, f x ∂μ = ⨍⁻ (x : α) in t, f x ∂μ

theorem MeasureTheory.setLaverage_congr_fun {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ENNReal} {g : α → ENNReal} (hs : MeasurableSet s) (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.laverage_lt_top {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤) : ⨍⁻ (x : α), f x ∂μ < ⊤

theorem MeasureTheory.setLaverage_lt_top {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ENNReal} : ∫⁻ (x : α) in s, f x ∂μ ≠ ⊤ → ⨍⁻ (x : α) in s, f x ∂μ < ⊤

theorem MeasureTheory.laverage_add_measure {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} {f : α → ENNReal} : ⨍⁻ (x : α), f x ∂(μ + ν) = ↑↑μ Set.univ / (↑↑μ Set.univ + ↑↑ν Set.univ) * ⨍⁻ (x : α), f x ∂μ + ↑↑ν Set.univ / (↑↑μ Set.univ + ↑↑ν Set.univ) * ⨍⁻ (x : α), f x ∂ν

theorem MeasureTheory.measure_mul_setLaverage {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (f : α → ENNReal) (h : ↑↑μ s ≠ ⊤) : ↑↑μ s * ⨍⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, f x ∂μ

theorem MeasureTheory.laverage_union {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {t : Set α} {f : α → ENNReal} (hd : MeasureTheory.AEDisjoint μ s t) (ht : MeasureTheory.NullMeasurableSet t) : ⨍⁻ (x : α) in s ∪ t, f x ∂μ = ↑↑μ s / (↑↑μ s + ↑↑μ t) * ⨍⁻ (x : α) in s, f x ∂μ + ↑↑μ t / (↑↑μ s + ↑↑μ t) * ⨍⁻ (x : α) in t, f x ∂μ

theorem MeasureTheory.laverage_union_mem_openSegment {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {t : Set α} {f : α → ENNReal} (hd : MeasureTheory.AEDisjoint μ s t) (ht : MeasureTheory.NullMeasurableSet t) (hs₀ : ↑↑μ s ≠ 0) (ht₀ : ↑↑μ t ≠ 0) (hsμ : ↑↑μ s ≠ ⊤) (htμ : ↑↑μ t ≠ ⊤) : ⨍⁻ (x : α) in s ∪ t, f x ∂μ ∈ openSegment ENNReal (⨍⁻ (x : α) in s, f x ∂μ) (⨍⁻ (x : α) in t, f x ∂μ)

theorem MeasureTheory.laverage_union_mem_segment {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {t : Set α} {f : α → ENNReal} (hd : MeasureTheory.AEDisjoint μ s t) (ht : MeasureTheory.NullMeasurableSet t) (hsμ : ↑↑μ s ≠ ⊤) (htμ : ↑↑μ t ≠ ⊤) : ⨍⁻ (x : α) in s ∪ t, f x ∂μ ∈ segment ENNReal (⨍⁻ (x : α) in s, f x ∂μ) (⨍⁻ (x : α) in t, f x ∂μ)

theorem MeasureTheory.laverage_mem_openSegment_compl_self {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ENNReal} [MeasureTheory.IsFiniteMeasure μ] (hs : MeasureTheory.NullMeasurableSet s) (hs₀ : ↑↑μ s ≠ 0) (hsc₀ : ↑↑μ sᶜ ≠ 0) : ⨍⁻ (x : α), f x ∂μ ∈ openSegment ENNReal (⨍⁻ (x : α) in s, f x ∂μ) (⨍⁻ (x : α) in sᶜ, f x ∂μ)

@[simp]

theorem MeasureTheory.laverage_const {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [h : NeZero μ] (c : ENNReal) : ⨍⁻ (_x : α), c ∂μ = c

theorem MeasureTheory.setLaverage_const {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs₀ : ↑↑μ s ≠ 0) (hs : ↑↑μ s ≠ ⊤) (c : ENNReal) : ⨍⁻ (_x : α) in s, c ∂μ = c

theorem MeasureTheory.laverage_one {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] [NeZero μ] : ⨍⁻ (_x : α), 1 ∂μ = 1

theorem MeasureTheory.setLaverage_one {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs₀ : ↑↑μ s ≠ 0) (hs : ↑↑μ s ≠ ⊤) : ⨍⁻ (_x : α) in s, 1 ∂μ = 1

theorem MeasureTheory.lintegral_laverage {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → ENNReal) : ∫⁻ (_x : α), ⨍⁻ (a : α), f a ∂μ ∂μ = ∫⁻ (x : α), f x ∂μ

theorem MeasureTheory.setLintegral_setLaverage {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → ENNReal) (s : Set α) : ∫⁻ (_x : α) in s, ⨍⁻ (a : α) in s, f a ∂μ ∂μ = ∫⁻ (x : α) in s, f x ∂μ

noncomputable def MeasureTheory.average {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) (f : α → E) : E

Average value of a function f w.r.t. a measure μ, notation: ⨍ x, f x ∂μ. It is defined as (μ univ).toReal⁻¹ • ∫ x, f x ∂μ, so it is equal to zero if f is not integrable or if μ is an infinite measure. If μ is a probability measure, then the average of any function is equal to its integral.

For the average on a set, use ⨍ x in s, f x ∂μ (defined as ⨍ x, f x ∂(μ.restrict s)). For average w.r.t. the volume, one can omit ∂volume.

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

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

Average value of a function f w.r.t. a measure μ, notation: ⨍ x, f x ∂μ. It is defined as (μ univ).toReal⁻¹ • ∫ x, f x ∂μ, so it is equal to zero if f is not integrable or if μ is an infinite measure. If μ is a probability measure, then the average of any function is equal to its integral.

For the average on a set, use ⨍ x in s, f x ∂μ (defined as ⨍ x, f x ∂(μ.restrict s)). For average w.r.t. the volume, one can omit ∂volume.

def MeasureTheory.«term⨍_,_» : Lean.ParserDescr

Average value of a function f w.r.t. a measure μ, notation: ⨍ x, f x ∂μ. It is defined as (μ univ).toReal⁻¹ • ∫ x, f x ∂μ, so it is equal to zero if f is not integrable or if μ is an infinite measure. If μ is a probability measure, then the average of any function is equal to its integral.

For the average on a set, use ⨍ x in s, f x ∂μ (defined as ⨍ x, f x ∂(μ.restrict s)). For average w.r.t. the volume, one can omit ∂volume.

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

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

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

Average value of a function f w.r.t. a measure μ, notation: ⨍ x, f x ∂μ. It is defined as (μ univ).toReal⁻¹ • ∫ x, f x ∂μ, so it is equal to zero if f is not integrable or if μ is an infinite measure. If μ is a probability measure, then the average of any function is equal to its integral.

For the average on a set, use ⨍ x in s, f x ∂μ (defined as ⨍ x, f x ∂(μ.restrict s)). For average w.r.t. the volume, one can omit ∂volume.

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

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

Average value of a function f w.r.t. a measure μ, notation: ⨍ x, f x ∂μ. It is defined as (μ univ).toReal⁻¹ • ∫ x, f x ∂μ, so it is equal to zero if f is not integrable or if μ is an infinite measure. If μ is a probability measure, then the average of any function is equal to its integral.

For the average on a set, use ⨍ x in s, f x ∂μ (defined as ⨍ x, f x ∂(μ.restrict s)). For average w.r.t. the volume, one can omit ∂volume.

@[simp]

theorem MeasureTheory.average_zero {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) : ⨍ (x : α), 0 ∂μ = 0

@[simp]

theorem MeasureTheory.average_zero_measure {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : α → E) : ⨍ (x : α), f x ∂0 = 0

@[simp]

theorem MeasureTheory.average_neg {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) (f : α → E) : ⨍ (x : α), -f x ∂μ = -⨍ (x : α), f x ∂μ

theorem MeasureTheory.average_eq' {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) (f : α → E) : ⨍ (x : α), f x ∂μ = ∫ (x : α), f x ∂(↑↑μ Set.univ)⁻¹ • μ

theorem MeasureTheory.average_eq {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) (f : α → E) : ⨍ (x : α), f x ∂μ = (ENNReal.toReal (↑↑μ Set.univ))⁻¹ • ∫ (x : α), f x ∂μ

theorem MeasureTheory.average_eq_integral {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] (f : α → E) : ⨍ (x : α), f x ∂μ = ∫ (x : α), f x ∂μ

@[simp]

theorem MeasureTheory.measure_smul_average {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → E) : ENNReal.toReal (↑↑μ Set.univ) • ⨍ (x : α), f x ∂μ = ∫ (x : α), f x ∂μ

theorem MeasureTheory.setAverage_eq {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) (f : α → E) (s : Set α) : ⨍ (x : α) in s, f x ∂μ = (ENNReal.toReal (↑↑μ s))⁻¹ • ∫ (x : α) in s, f x ∂μ

theorem MeasureTheory.setAverage_eq' {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α) (f : α → E) (s : Set α) : ⨍ (x : α) in s, f x ∂μ = ∫ (x : α), f x ∂(↑↑μ s)⁻¹ • MeasureTheory.Measure.restrict μ s

theorem MeasureTheory.average_congr {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} {f : α → E} {g : α → E} (h : f =ᶠ[MeasureTheory.Measure.ae μ] g) : ⨍ (x : α), f x ∂μ = ⨍ (x : α), g x ∂μ

theorem MeasureTheory.setAverage_congr {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} {s : Set α} {t : Set α} {f : α → E} (h : s =ᶠ[MeasureTheory.Measure.ae μ] t) : ⨍ (x : α) in s, f x ∂μ = ⨍ (x : α) in t, f x ∂μ

theorem MeasureTheory.setAverage_congr_fun {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} {s : Set α} {f : α → E} {g : α → E} (hs : MeasurableSet s) (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.average_add_measure {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure ν] {f : α → E} (hμ : MeasureTheory.Integrable f) (hν : MeasureTheory.Integrable f) : ⨍ (x : α), f x ∂(μ + ν) = (ENNReal.toReal (↑↑μ Set.univ) / (ENNReal.toReal (↑↑μ Set.univ) + ENNReal.toReal (↑↑ν Set.univ))) • ⨍ (x : α), f x ∂μ + (ENNReal.toReal (↑↑ν Set.univ) / (ENNReal.toReal (↑↑μ Set.univ) + ENNReal.toReal (↑↑ν Set.univ))) • ⨍ (x : α), f x ∂ν

theorem MeasureTheory.average_pair {α : Type u_1} {E : Type u_2} {F : Type u_3} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {μ : MeasureTheory.Measure α} {f : α → E} {g : α → F} (hfi : MeasureTheory.Integrable f) (hgi : MeasureTheory.Integrable g) : ⨍ (x : α), (f x, g x) ∂μ = (⨍ (x : α), f x ∂μ, ⨍ (x : α), g x ∂μ)

theorem MeasureTheory.measure_smul_setAverage {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} (f : α → E) {s : Set α} (h : ↑↑μ s ≠ ⊤) : ENNReal.toReal (↑↑μ s) • ⨍ (x : α) in s, f x ∂μ = ∫ (x : α) in s, f x ∂μ

theorem MeasureTheory.average_union {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} {f : α → E} {s : Set α} {t : Set α} (hd : MeasureTheory.AEDisjoint μ s t) (ht : MeasureTheory.NullMeasurableSet t) (hsμ : ↑↑μ s ≠ ⊤) (htμ : ↑↑μ t ≠ ⊤) (hfs : MeasureTheory.IntegrableOn f s) (hft : MeasureTheory.IntegrableOn f t) : ⨍ (x : α) in s ∪ t, f x ∂μ = (ENNReal.toReal (↑↑μ s) / (ENNReal.toReal (↑↑μ s) + ENNReal.toReal (↑↑μ t))) • ⨍ (x : α) in s, f x ∂μ + (ENNReal.toReal (↑↑μ t) / (ENNReal.toReal (↑↑μ s) + ENNReal.toReal (↑↑μ t))) • ⨍ (x : α) in t, f x ∂μ

theorem MeasureTheory.average_union_mem_openSegment {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} {f : α → E} {s : Set α} {t : Set α} (hd : MeasureTheory.AEDisjoint μ s t) (ht : MeasureTheory.NullMeasurableSet t) (hs₀ : ↑↑μ s ≠ 0) (ht₀ : ↑↑μ t ≠ 0) (hsμ : ↑↑μ s ≠ ⊤) (htμ : ↑↑μ t ≠ ⊤) (hfs : MeasureTheory.IntegrableOn f s) (hft : MeasureTheory.IntegrableOn f t) : ⨍ (x : α) in s ∪ t, f x ∂μ ∈ openSegment ℝ (⨍ (x : α) in s, f x ∂μ) (⨍ (x : α) in t, f x ∂μ)

theorem MeasureTheory.average_union_mem_segment {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} {f : α → E} {s : Set α} {t : Set α} (hd : MeasureTheory.AEDisjoint μ s t) (ht : MeasureTheory.NullMeasurableSet t) (hsμ : ↑↑μ s ≠ ⊤) (htμ : ↑↑μ t ≠ ⊤) (hfs : MeasureTheory.IntegrableOn f s) (hft : MeasureTheory.IntegrableOn f t) : ⨍ (x : α) in s ∪ t, f x ∂μ ∈ segment ℝ (⨍ (x : α) in s, f x ∂μ) (⨍ (x : α) in t, f x ∂μ)

theorem MeasureTheory.average_mem_openSegment_compl_self {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {f : α → E} {s : Set α} (hs : MeasureTheory.NullMeasurableSet s) (hs₀ : ↑↑μ s ≠ 0) (hsc₀ : ↑↑μ sᶜ ≠ 0) (hfi : MeasureTheory.Integrable f) : ⨍ (x : α), f x ∂μ ∈ openSegment ℝ (⨍ (x : α) in s, f x ∂μ) (⨍ (x : α) in sᶜ, f x ∂μ)

@[simp]

theorem MeasureTheory.average_const {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [h : NeZero μ] (c : E) : ⨍ (_x : α), c ∂μ = c

theorem MeasureTheory.setAverage_const {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {s : Set α} (hs₀ : ↑↑μ s ≠ 0) (hs : ↑↑μ s ≠ ⊤) (c : E) : ⨍ (x : α) in s, c ∂μ = c

theorem MeasureTheory.integral_average {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → E) : ∫ (x : α), ⨍ (a : α), f a ∂μ ∂μ = ∫ (x : α), f x ∂μ

theorem MeasureTheory.setIntegral_setAverage {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → E) (s : Set α) : ∫ (x : α) in s, ⨍ (a : α) in s, f a ∂μ ∂μ = ∫ (x : α) in s, f x ∂μ

theorem MeasureTheory.integral_sub_average {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → E) : ∫ (x : α), f x - ⨍ (a : α), f a ∂μ ∂μ = 0

theorem MeasureTheory.setAverage_sub_setAverage {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {s : Set α} (hs : ↑↑μ s ≠ ⊤) (f : α → E) : ∫ (x : α) in s, f x - ⨍ (a : α) in s, f a ∂μ ∂μ = 0

theorem MeasureTheory.integral_average_sub {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {f : α → E} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Integrable f) : ∫ (x : α), ⨍ (a : α), f a ∂μ - f x ∂μ = 0

theorem MeasureTheory.setIntegral_setAverage_sub {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {s : Set α} {f : α → E} (hs : ↑↑μ s ≠ ⊤) (hf : MeasureTheory.IntegrableOn f s) : ∫ (x : α) in s, ⨍ (a : α) in s, f a ∂μ - f x ∂μ = 0

theorem MeasureTheory.ofReal_average {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable f) (hf₀ : 0 ≤ᶠ[MeasureTheory.Measure.ae μ] f) : ENNReal.ofReal (⨍ (x : α), f x ∂μ) = (∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ) / ↑↑μ Set.univ

theorem MeasureTheory.ofReal_setAverage {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ} (hf : MeasureTheory.IntegrableOn f s) (hf₀ : 0 ≤ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.restrict μ s)] f) : ENNReal.ofReal (⨍ (x : α) in s, f x ∂μ) = (∫⁻ (x : α) in s, ENNReal.ofReal (f x) ∂μ) / ↑↑μ s

theorem MeasureTheory.toReal_laverage {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hf : AEMeasurable f) (hf' : ∀ᵐ (x : α) ∂μ, f x ≠ ⊤) : ENNReal.toReal (⨍⁻ (x : α), f x ∂μ) = ⨍ (x : α), ENNReal.toReal (f x) ∂μ

theorem MeasureTheory.toReal_setLaverage {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ENNReal} (hf : AEMeasurable f) (hf' : ∀ᵐ (x : α) ∂MeasureTheory.Measure.restrict μ s, f x ≠ ⊤) : ENNReal.toReal (⨍⁻ (x : α) in s, f x ∂μ) = ⨍ (x : α) in s, ENNReal.toReal (f x) ∂μ

First moment method

theorem MeasureTheory.measure_le_setAverage_pos {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ} (hμ : ↑↑μ s ≠ 0) (hμ₁ : ↑↑μ s ≠ ⊤) (hf : MeasureTheory.IntegrableOn f s) : 0 < ↑↑μ {x | x ∈ s ∧ f x ≤ ⨍ (a : α) in s, f a ∂μ}

First moment method. An integrable function is smaller than its mean on a set of positive measure.

theorem MeasureTheory.measure_setAverage_le_pos {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ} (hμ : ↑↑μ s ≠ 0) (hμ₁ : ↑↑μ s ≠ ⊤) (hf : MeasureTheory.IntegrableOn f s) : 0 < ↑↑μ {x | x ∈ s ∧ ⨍ (a : α) in s, f a ∂μ ≤ f x}

First moment method. An integrable function is greater than its mean on a set of positive measure.

theorem MeasureTheory.exists_le_setAverage {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ} (hμ : ↑↑μ s ≠ 0) (hμ₁ : ↑↑μ s ≠ ⊤) (hf : MeasureTheory.IntegrableOn f s) : ∃ x, x ∈ s ∧ f x ≤ ⨍ (a : α) in s, f a ∂μ

First moment method. The minimum of an integrable function is smaller than its mean.

theorem MeasureTheory.exists_setAverage_le {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ} (hμ : ↑↑μ s ≠ 0) (hμ₁ : ↑↑μ s ≠ ⊤) (hf : MeasureTheory.IntegrableOn f s) : ∃ x, x ∈ s ∧ ⨍ (a : α) in s, f a ∂μ ≤ f x

First moment method. The maximum of an integrable function is greater than its mean.

theorem MeasureTheory.measure_le_average_pos {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : MeasureTheory.Integrable f) : 0 < ↑↑μ {x | f x ≤ ⨍ (a : α), f a ∂μ}

First moment method. An integrable function is smaller than its mean on a set of positive measure.

theorem MeasureTheory.measure_average_le_pos {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : MeasureTheory.Integrable f) : 0 < ↑↑μ {x | ⨍ (a : α), f a ∂μ ≤ f x}

First moment method. An integrable function is greater than its mean on a set of positive measure.

theorem MeasureTheory.exists_le_average {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : MeasureTheory.Integrable f) : ∃ x, f x ≤ ⨍ (a : α), f a ∂μ

First moment method. The minimum of an integrable function is smaller than its mean.

theorem MeasureTheory.exists_average_le {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : MeasureTheory.Integrable f) : ∃ x, ⨍ (a : α), f a ∂μ ≤ f x

First moment method. The maximum of an integrable function is greater than its mean.

theorem MeasureTheory.exists_not_mem_null_le_average {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : MeasureTheory.Integrable f) (hN : ↑↑μ N = 0) : ∃ x, ¬x ∈ N ∧ f x ≤ ⨍ (a : α), f a ∂μ

First moment method. The minimum of an integrable function is smaller than its mean, while avoiding a null set.

theorem MeasureTheory.exists_not_mem_null_average_le {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : MeasureTheory.Integrable f) (hN : ↑↑μ N = 0) : ∃ x, ¬x ∈ N ∧ ⨍ (a : α), f a ∂μ ≤ f x

First moment method. The maximum of an integrable function is greater than its mean, while avoiding a null set.

theorem MeasureTheory.measure_le_integral_pos {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsProbabilityMeasure μ] (hf : MeasureTheory.Integrable f) : 0 < ↑↑μ {x | f x ≤ ∫ (a : α), f a ∂μ}

First moment method. An integrable function is smaller than its integral on a set of positive measure.

theorem MeasureTheory.measure_integral_le_pos {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsProbabilityMeasure μ] (hf : MeasureTheory.Integrable f) : 0 < ↑↑μ {x | ∫ (a : α), f a ∂μ ≤ f x}

First moment method. An integrable function is greater than its integral on a set of positive measure.

theorem MeasureTheory.exists_le_integral {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsProbabilityMeasure μ] (hf : MeasureTheory.Integrable f) : ∃ x, f x ≤ ∫ (a : α), f a ∂μ

First moment method. The minimum of an integrable function is smaller than its integral.

theorem MeasureTheory.exists_integral_le {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsProbabilityMeasure μ] (hf : MeasureTheory.Integrable f) : ∃ x, ∫ (a : α), f a ∂μ ≤ f x

First moment method. The maximum of an integrable function is greater than its integral.

theorem MeasureTheory.exists_not_mem_null_le_integral {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ℝ} [MeasureTheory.IsProbabilityMeasure μ] (hf : MeasureTheory.Integrable f) (hN : ↑↑μ N = 0) : ∃ x, ¬x ∈ N ∧ f x ≤ ∫ (a : α), f a ∂μ

First moment method. The minimum of an integrable function is smaller than its integral, while avoiding a null set.

theorem MeasureTheory.exists_not_mem_null_integral_le {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ℝ} [MeasureTheory.IsProbabilityMeasure μ] (hf : MeasureTheory.Integrable f) (hN : ↑↑μ N = 0) : ∃ x, ¬x ∈ N ∧ ∫ (a : α), f a ∂μ ≤ f x

First moment method. The maximum of an integrable function is greater than its integral, while avoiding a null set.

theorem MeasureTheory.measure_le_setLaverage_pos {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ENNReal} (hμ : ↑↑μ s ≠ 0) (hμ₁ : ↑↑μ s ≠ ⊤) (hf : AEMeasurable f) : 0 < ↑↑μ {x | x ∈ s ∧ f x ≤ ⨍⁻ (a : α) in s, f a ∂μ}

First moment method. A measurable function is smaller than its mean on a set of positive measure.

theorem MeasureTheory.measure_setLaverage_le_pos {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ENNReal} (hμ : ↑↑μ s ≠ 0) (hs : MeasureTheory.NullMeasurableSet s) (hint : ∫⁻ (a : α) in s, f a ∂μ ≠ ⊤) : 0 < ↑↑μ {x | x ∈ s ∧ ⨍⁻ (a : α) in s, f a ∂μ ≤ f x}

First moment method. A measurable function is greater than its mean on a set of positive measure.

theorem MeasureTheory.exists_le_setLaverage {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ENNReal} (hμ : ↑↑μ s ≠ 0) (hμ₁ : ↑↑μ s ≠ ⊤) (hf : AEMeasurable f) : ∃ x, x ∈ s ∧ f x ≤ ⨍⁻ (a : α) in s, f a ∂μ

First moment method. The minimum of a measurable function is smaller than its mean.

theorem MeasureTheory.exists_setLaverage_le {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ENNReal} (hμ : ↑↑μ s ≠ 0) (hs : MeasureTheory.NullMeasurableSet s) (hint : ∫⁻ (a : α) in s, f a ∂μ ≠ ⊤) : ∃ x, x ∈ s ∧ ⨍⁻ (a : α) in s, f a ∂μ ≤ f x

First moment method. The maximum of a measurable function is greater than its mean.

theorem MeasureTheory.measure_laverage_le_pos {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hμ : μ ≠ 0) (hint : ∫⁻ (a : α), f a ∂μ ≠ ⊤) : 0 < ↑↑μ {x | ⨍⁻ (a : α), f a ∂μ ≤ f x}

First moment method. A measurable function is greater than its mean on a set of positive measure.

theorem MeasureTheory.exists_laverage_le {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} (hμ : μ ≠ 0) (hint : ∫⁻ (a : α), f a ∂μ ≠ ⊤) : ∃ x, ⨍⁻ (a : α), f a ∂μ ≤ f x

First moment method. The maximum of a measurable function is greater than its mean.

theorem MeasureTheory.exists_not_mem_null_laverage_le {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ENNReal} (hμ : μ ≠ 0) (hint : ∫⁻ (a : α), f a ∂μ ≠ ⊤) (hN : ↑↑μ N = 0) : ∃ x, ¬x ∈ N ∧ ⨍⁻ (a : α), f a ∂μ ≤ f x

First moment method. The maximum of a measurable function is greater than its mean, while avoiding a null set.

theorem MeasureTheory.measure_le_laverage_pos {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : AEMeasurable f) : 0 < ↑↑μ {x | f x ≤ ⨍⁻ (a : α), f a ∂μ}

First moment method. A measurable function is smaller than its mean on a set of positive measure.

theorem MeasureTheory.exists_le_laverage {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : AEMeasurable f) : ∃ x, f x ≤ ⨍⁻ (a : α), f a ∂μ

First moment method. The minimum of a measurable function is smaller than its mean.

theorem MeasureTheory.exists_not_mem_null_le_laverage {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ENNReal} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (hf : AEMeasurable f) (hN : ↑↑μ N = 0) : ∃ x, ¬x ∈ N ∧ f x ≤ ⨍⁻ (a : α), f a ∂μ

First moment method. The minimum of a measurable function is smaller than its mean, while avoiding a null set.

theorem MeasureTheory.measure_le_lintegral_pos {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} [MeasureTheory.IsProbabilityMeasure μ] (hf : AEMeasurable f) : 0 < ↑↑μ {x | f x ≤ ∫⁻ (a : α), f a ∂μ}

First moment method. A measurable function is smaller than its integral on a set f positive measure.

theorem MeasureTheory.measure_lintegral_le_pos {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} [MeasureTheory.IsProbabilityMeasure μ] (hint : ∫⁻ (a : α), f a ∂μ ≠ ⊤) : 0 < ↑↑μ {x | ∫⁻ (a : α), f a ∂μ ≤ f x}

First moment method. A measurable function is greater than its integral on a set f positive measure.

theorem MeasureTheory.exists_le_lintegral {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} [MeasureTheory.IsProbabilityMeasure μ] (hf : AEMeasurable f) : ∃ x, f x ≤ ∫⁻ (a : α), f a ∂μ

First moment method. The minimum of a measurable function is smaller than its integral.

theorem MeasureTheory.exists_lintegral_le {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} [MeasureTheory.IsProbabilityMeasure μ] (hint : ∫⁻ (a : α), f a ∂μ ≠ ⊤) : ∃ x, ∫⁻ (a : α), f a ∂μ ≤ f x

First moment method. The maximum of a measurable function is greater than its integral.

theorem MeasureTheory.exists_not_mem_null_le_lintegral {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ENNReal} [MeasureTheory.IsProbabilityMeasure μ] (hf : AEMeasurable f) (hN : ↑↑μ N = 0) : ∃ x, ¬x ∈ N ∧ f x ≤ ∫⁻ (a : α), f a ∂μ

First moment method. The minimum of a measurable function is smaller than its integral, while avoiding a null set.

theorem MeasureTheory.exists_not_mem_null_lintegral_le {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ENNReal} [MeasureTheory.IsProbabilityMeasure μ] (hint : ∫⁻ (a : α), f a ∂μ ≠ ⊤) (hN : ↑↑μ N = 0) : ∃ x, ¬x ∈ N ∧ ∫⁻ (a : α), f a ∂μ ≤ f x

First moment method. The maximum of a measurable function is greater than its integral, while avoiding a null set.

theorem MeasureTheory.tendsto_integral_smul_of_tendsto_average_norm_sub {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {ι : Type u_4} {a : ι → Set α} {l : Filter ι} {f : α → E} {c : E} {g : ι → α → ℝ} (K : ℝ) (hf : Filter.Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (nhds 0)) (f_int : ∀ᶠ (i : ι) in l, MeasureTheory.IntegrableOn f (a i)) (hg : Filter.Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (nhds 1)) (g_supp : ∀ᶠ (i : ι) in l, Function.support (g i) ⊆ a i) (g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / ENNReal.toReal (↑↑μ (a i))) : Filter.Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (nhds c)

If the average of a function f along a sequence of sets aₙ converges to c (more precisely, we require that ⨍ y in a i, ‖f y - c‖ ∂μ tends to 0), then the integral of gₙ • f also tends to c if gₙ is supported in aₙ, has integral converging to one and supremum at most K / μ aₙ.