# 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 : } (μ : ) (f : αENNReal) :

Average value of an ℝ≥0∞-valued function f w.r.t. a measure μ, denoted ⨍⁻ x, f x ∂μ.

It is equal to (μ univ)⁻¹ * ∫⁻ x, f x ∂μ, so it takes value zero if μ is an infinite measure. If μ is a probability measure, then the average of any function is equal to its integral.

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

Equations
• = ∫⁻ (x : α), f x(μ Set.univ)⁻¹ μ
Instances For

Pretty printer defined by notation3 command.

Equations
• One or more equations did not get rendered due to their size.
Instances For

Average value of an ℝ≥0∞-valued function f w.r.t. a measure μ.

It is equal to (μ univ)⁻¹ * ∫⁻ x, f x ∂μ, so it takes value zero if μ is an infinite measure. If μ is a probability measure, then the average of any function is equal to its integral.

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

Equations
• One or more equations did not get rendered due to their size.
Instances For

Average value of an ℝ≥0∞-valued function f w.r.t. to the standard measure.

It is equal to (volume univ)⁻¹ * ∫⁻ x, f x, so it takes value zero if the space has infinite measure. In a probability space, the average of any function is equal to its integral.

For the average on a set, use ⨍⁻ x in s, f x, defined as ⨍⁻ x, f x ∂(volume.restrict s).

Equations
• One or more equations did not get rendered due to their size.
Instances For

Pretty printer defined by notation3 command.

Equations
• One or more equations did not get rendered due to their size.
Instances For

Pretty printer defined by notation3 command.

Equations
• One or more equations did not get rendered due to their size.
Instances For

Average value of an ℝ≥0∞-valued function f w.r.t. a measure μ on a set s.

It is equal to (μ s)⁻¹ * ∫⁻ x, f x ∂μ, so it takes value zero if s has infinite measure. If s has measure 1, then the average of any function is equal to its integral.

For the average w.r.t. the volume, one can omit ∂volume.

Equations
• One or more equations did not get rendered due to their size.
Instances For

Average value of an ℝ≥0∞-valued function f w.r.t. to the standard measure on a set s.

It is equal to (volume s)⁻¹ * ∫⁻ x, f x, so it takes value zero if s has infinite measure. If s has measure 1, then the average of any function is equal to its integral.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem MeasureTheory.laverage_zero {α : Type u_1} {m0 : } (μ : ) :
⨍⁻ (_x : α), 0μ = 0
@[simp]
theorem MeasureTheory.laverage_zero_measure {α : Type u_1} {m0 : } (f : αENNReal) :
⨍⁻ (x : α), f x0 = 0
theorem MeasureTheory.laverage_eq' {α : Type u_1} {m0 : } (μ : ) (f : αENNReal) :
⨍⁻ (x : α), f xμ = ∫⁻ (x : α), f x(μ Set.univ)⁻¹ μ
theorem MeasureTheory.laverage_eq {α : Type u_1} {m0 : } (μ : ) (f : αENNReal) :
⨍⁻ (x : α), f xμ = (∫⁻ (x : α), f xμ) / μ Set.univ
theorem MeasureTheory.laverage_eq_lintegral {α : Type u_1} {m0 : } (μ : ) (f : αENNReal) :
⨍⁻ (x : α), f xμ = ∫⁻ (x : α), f xμ
@[simp]
theorem MeasureTheory.measure_mul_laverage {α : Type u_1} {m0 : } (μ : ) (f : αENNReal) :
μ Set.univ * ⨍⁻ (x : α), f xμ = ∫⁻ (x : α), f xμ
theorem MeasureTheory.setLaverage_eq {α : Type u_1} {m0 : } (μ : ) (f : αENNReal) (s : Set α) :
⨍⁻ (x : α) in s, f xμ = (∫⁻ (x : α) in s, f xμ) / μ s
theorem MeasureTheory.setLaverage_eq' {α : Type u_1} {m0 : } (μ : ) (f : αENNReal) (s : Set α) :
⨍⁻ (x : α) in s, f xμ = ∫⁻ (x : α), f x(μ s)⁻¹ μ.restrict s
theorem MeasureTheory.laverage_congr {α : Type u_1} {m0 : } {μ : } {f : αENNReal} {g : αENNReal} (h : f =ᶠ[μ.ae] g) :
⨍⁻ (x : α), f xμ = ⨍⁻ (x : α), g xμ
theorem MeasureTheory.setLaverage_congr {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} {f : αENNReal} (h : s =ᶠ[μ.ae] t) :
⨍⁻ (x : α) in s, f xμ = ⨍⁻ (x : α) in t, f xμ
theorem MeasureTheory.setLaverage_congr_fun {α : Type u_1} {m0 : } {μ : } {s : Set α} {f : αENNReal} {g : αENNReal} (hs : ) (h : ∀ᵐ (x : α) ∂μ, x sf x = g x) :
⨍⁻ (x : α) in s, f xμ = ⨍⁻ (x : α) in s, g xμ
theorem MeasureTheory.laverage_lt_top {α : Type u_1} {m0 : } {μ : } {f : αENNReal} (hf : ∫⁻ (x : α), f xμ ) :
⨍⁻ (x : α), f xμ <
theorem MeasureTheory.setLaverage_lt_top {α : Type u_1} {m0 : } {μ : } {s : Set α} {f : αENNReal} :
∫⁻ (x : α) in s, f xμ ⨍⁻ (x : α) in s, f xμ <
theorem MeasureTheory.laverage_add_measure {α : Type u_1} {m0 : } {μ : } {ν : } {f : αENNReal} :
⨍⁻ (x : α), f x(μ + ν) = μ Set.univ / (μ Set.univ + ν Set.univ) * ⨍⁻ (x : α), f xμ + ν Set.univ / (μ Set.univ + ν Set.univ) * ⨍⁻ (x : α), f xν
theorem MeasureTheory.measure_mul_setLaverage {α : Type u_1} {m0 : } {μ : } {s : Set α} (f : αENNReal) (h : μ s ) :
μ s * ⨍⁻ (x : α) in s, f xμ = ∫⁻ (x : α) in s, f xμ
theorem MeasureTheory.laverage_union {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} {f : αENNReal} (hd : ) (ht : ) :
⨍⁻ (x : α) in s t, f xμ = μ s / (μ s + μ t) * ⨍⁻ (x : α) in s, f xμ + μ t / (μ s + μ t) * ⨍⁻ (x : α) in t, f xμ
theorem MeasureTheory.laverage_union_mem_openSegment {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} {f : αENNReal} (hd : ) (ht : ) (hs₀ : μ s 0) (ht₀ : μ t 0) (hsμ : μ s ) (htμ : μ t ) :
⨍⁻ (x : α) in s t, f xμ openSegment ENNReal (⨍⁻ (x : α) in s, f xμ) (⨍⁻ (x : α) in t, f xμ)
theorem MeasureTheory.laverage_union_mem_segment {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} {f : αENNReal} (hd : ) (ht : ) (hsμ : μ s ) (htμ : μ t ) :
⨍⁻ (x : α) in s t, f xμ segment ENNReal (⨍⁻ (x : α) in s, f xμ) (⨍⁻ (x : α) in t, f xμ)
theorem MeasureTheory.laverage_mem_openSegment_compl_self {α : Type u_1} {m0 : } {μ : } {s : Set α} {f : αENNReal} (hs : ) (hs₀ : μ s 0) (hsc₀ : μ s 0) :
⨍⁻ (x : α), f xμ openSegment ENNReal (⨍⁻ (x : α) in s, f xμ) (⨍⁻ (x : α) in s, f xμ)
@[simp]
theorem MeasureTheory.laverage_const {α : Type u_1} {m0 : } (μ : ) [h : ] (c : ENNReal) :
⨍⁻ (_x : α), cμ = c
theorem MeasureTheory.setLaverage_const {α : Type u_1} {m0 : } {μ : } {s : Set α} (hs₀ : μ s 0) (hs : μ s ) (c : ENNReal) :
⨍⁻ (_x : α) in s, cμ = c
theorem MeasureTheory.laverage_one {α : Type u_1} {m0 : } {μ : } [] :
⨍⁻ (_x : α), 1μ = 1
theorem MeasureTheory.setLaverage_one {α : Type u_1} {m0 : } {μ : } {s : Set α} (hs₀ : μ s 0) (hs : μ s ) :
⨍⁻ (_x : α) in s, 1μ = 1
theorem MeasureTheory.lintegral_laverage {α : Type u_1} {m0 : } (μ : ) (f : αENNReal) :
∫⁻ (_x : α), ⨍⁻ (a : α), f aμμ = ∫⁻ (x : α), f xμ
theorem MeasureTheory.setLintegral_setLaverage {α : Type u_1} {m0 : } (μ : ) (f : αENNReal) (s : Set α) :
∫⁻ (_x : α) in s, ⨍⁻ (a : α) in s, f aμμ = ∫⁻ (x : α) in s, f xμ
noncomputable def MeasureTheory.average {α : Type u_1} {E : Type u_2} {m0 : } [] (μ : ) (f : αE) :
E

Average value of a function f w.r.t. a measure μ, denoted ⨍ x, f x ∂μ.

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

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

Equations
• = ∫ (x : α), f x(μ Set.univ)⁻¹ μ
Instances For

Pretty printer defined by notation3 command.

Equations
• One or more equations did not get rendered due to their size.
Instances For

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

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

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

Equations
• One or more equations did not get rendered due to their size.
Instances For

Average value of a function f w.r.t. to the standard measure.

It is equal to (volume univ).toReal⁻¹ * ∫ x, f x, so it takes value zero if f is not integrable or if the space has infinite measure. In a probability space, the average of any function is equal to its integral.

For the average on a set, use ⨍ x in s, f x, defined as ⨍ x, f x ∂(volume.restrict s).

Equations
• One or more equations did not get rendered due to their size.
Instances For

Pretty printer defined by notation3 command.

Equations
• One or more equations did not get rendered due to their size.
Instances For

Pretty printer defined by notation3 command.

Equations
• One or more equations did not get rendered due to their size.
Instances For

Average value of a function f w.r.t. a measure μ on a set s.

It is equal to (μ s).toReal⁻¹ * ∫ x, f x ∂μ, so it takes value zero if f is not integrable on s or if s has infinite measure. If s has measure 1, then the average of any function is equal to its integral.

For the average w.r.t. the volume, one can omit ∂volume.

Equations
• One or more equations did not get rendered due to their size.
Instances For

Pretty printer defined by notation3 command.

Equations
• One or more equations did not get rendered due to their size.
Instances For

Average value of a function f w.r.t. to the standard measure on a set s.

It is equal to (volume s).toReal⁻¹ * ∫ x, f x, so it takes value zero f is not integrable on s or if s has infinite measure. If s has measure 1, then the average of any function is equal to its integral.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem MeasureTheory.average_zero {α : Type u_1} {E : Type u_2} {m0 : } [] (μ : ) :
⨍ (x : α), 0μ = 0
@[simp]
theorem MeasureTheory.average_zero_measure {α : Type u_1} {E : Type u_2} {m0 : } [] (f : αE) :
⨍ (x : α), f x0 = 0
@[simp]
theorem MeasureTheory.average_neg {α : Type u_1} {E : Type u_2} {m0 : } [] (μ : ) (f : αE) :
⨍ (x : α), -f xμ = -⨍ (x : α), f xμ
theorem MeasureTheory.average_eq' {α : Type u_1} {E : Type u_2} {m0 : } [] (μ : ) (f : αE) :
⨍ (x : α), f xμ = ∫ (x : α), f x(μ Set.univ)⁻¹ μ
theorem MeasureTheory.average_eq {α : Type u_1} {E : Type u_2} {m0 : } [] (μ : ) (f : αE) :
⨍ (x : α), f xμ = (μ Set.univ).toReal⁻¹ ∫ (x : α), f xμ
theorem MeasureTheory.average_eq_integral {α : Type u_1} {E : Type u_2} {m0 : } [] (μ : ) (f : αE) :
⨍ (x : α), f xμ = ∫ (x : α), f xμ
@[simp]
theorem MeasureTheory.measure_smul_average {α : Type u_1} {E : Type u_2} {m0 : } [] (μ : ) (f : αE) :
(μ Set.univ).toReal ⨍ (x : α), f xμ = ∫ (x : α), f xμ
theorem MeasureTheory.setAverage_eq {α : Type u_1} {E : Type u_2} {m0 : } [] (μ : ) (f : αE) (s : Set α) :
⨍ (x : α) in s, f xμ = (μ s).toReal⁻¹ ∫ (x : α) in s, f xμ
theorem MeasureTheory.setAverage_eq' {α : Type u_1} {E : Type u_2} {m0 : } [] (μ : ) (f : αE) (s : Set α) :
⨍ (x : α) in s, f xμ = ∫ (x : α), f x(μ s)⁻¹ μ.restrict s
theorem MeasureTheory.average_congr {α : Type u_1} {E : Type u_2} {m0 : } [] {μ : } {f : αE} {g : αE} (h : f =ᶠ[μ.ae] g) :
⨍ (x : α), f xμ = ⨍ (x : α), g xμ
theorem MeasureTheory.setAverage_congr {α : Type u_1} {E : Type u_2} {m0 : } [] {μ : } {s : Set α} {t : Set α} {f : αE} (h : s =ᶠ[μ.ae] t) :
⨍ (x : α) in s, f xμ = ⨍ (x : α) in t, f xμ
theorem MeasureTheory.setAverage_congr_fun {α : Type u_1} {E : Type u_2} {m0 : } [] {μ : } {s : Set α} {f : αE} {g : αE} (hs : ) (h : ∀ᵐ (x : α) ∂μ, x sf x = g x) :
⨍ (x : α) in s, f xμ = ⨍ (x : α) in s, g xμ
theorem MeasureTheory.average_add_measure {α : Type u_1} {E : Type u_2} {m0 : } [] {μ : } {ν : } {f : αE} (hμ : ) (hν : ) :
⨍ (x : α), f x(μ + ν) = ((μ Set.univ).toReal / ((μ Set.univ).toReal + (ν Set.univ).toReal)) ⨍ (x : α), f xμ + ((ν Set.univ).toReal / ((μ Set.univ).toReal + (ν Set.univ).toReal)) ⨍ (x : α), f xν
theorem MeasureTheory.average_pair {α : Type u_1} {E : Type u_2} {F : Type u_3} {m0 : } [] [] [] [] {μ : } {f : αE} {g : αF} (hfi : ) (hgi : ) :
⨍ (x : α), (f x, g x)μ = (⨍ (x : α), f xμ, ⨍ (x : α), g xμ)
theorem MeasureTheory.measure_smul_setAverage {α : Type u_1} {E : Type u_2} {m0 : } [] {μ : } (f : αE) {s : Set α} (h : μ s ) :
(μ s).toReal ⨍ (x : α) in s, f xμ = ∫ (x : α) in s, f xμ
theorem MeasureTheory.average_union {α : Type u_1} {E : Type u_2} {m0 : } [] {μ : } {f : αE} {s : Set α} {t : Set α} (hd : ) (ht : ) (hsμ : μ s ) (htμ : μ t ) (hfs : ) (hft : ) :
⨍ (x : α) in s t, f xμ = ((μ s).toReal / ((μ s).toReal + (μ t).toReal)) ⨍ (x : α) in s, f xμ + ((μ t).toReal / ((μ s).toReal + (μ t).toReal)) ⨍ (x : α) in t, f xμ
theorem MeasureTheory.average_union_mem_openSegment {α : Type u_1} {E : Type u_2} {m0 : } [] {μ : } {f : αE} {s : Set α} {t : Set α} (hd : ) (ht : ) (hs₀ : μ s 0) (ht₀ : μ t 0) (hsμ : μ s ) (htμ : μ t ) (hfs : ) (hft : ) :
⨍ (x : α) in s t, f xμ openSegment (⨍ (x : α) in s, f xμ) (⨍ (x : α) in t, f xμ)
theorem MeasureTheory.average_union_mem_segment {α : Type u_1} {E : Type u_2} {m0 : } [] {μ : } {f : αE} {s : Set α} {t : Set α} (hd : ) (ht : ) (hsμ : μ s ) (htμ : μ t ) (hfs : ) (hft : ) :
⨍ (x : α) in s t, f xμ segment (⨍ (x : α) in s, f xμ) (⨍ (x : α) in t, f xμ)
theorem MeasureTheory.average_mem_openSegment_compl_self {α : Type u_1} {E : Type u_2} {m0 : } [] {μ : } {f : αE} {s : Set α} (hs : ) (hs₀ : μ s 0) (hsc₀ : μ s 0) (hfi : ) :
⨍ (x : α), f xμ openSegment (⨍ (x : α) in s, f xμ) (⨍ (x : α) in s, f xμ)
@[simp]
theorem MeasureTheory.average_const {α : Type u_1} {E : Type u_2} {m0 : } [] [] (μ : ) [h : ] (c : E) :
⨍ (_x : α), cμ = c
theorem MeasureTheory.setAverage_const {α : Type u_1} {E : Type u_2} {m0 : } [] [] {μ : } {s : Set α} (hs₀ : μ s 0) (hs : μ s ) (c : E) :
⨍ (x : α) in s, cμ = c
theorem MeasureTheory.integral_average {α : Type u_1} {E : Type u_2} {m0 : } [] [] (μ : ) (f : αE) :
∫ (x : α), ⨍ (a : α), f aμμ = ∫ (x : α), f xμ
theorem MeasureTheory.setIntegral_setAverage {α : Type u_1} {E : Type u_2} {m0 : } [] [] (μ : ) (f : αE) (s : Set α) :
∫ (x : α) in s, ⨍ (a : α) in s, f aμμ = ∫ (x : α) in s, f xμ
theorem MeasureTheory.integral_sub_average {α : Type u_1} {E : Type u_2} {m0 : } [] [] (μ : ) (f : αE) :
∫ (x : α), f x - ⨍ (a : α), f aμμ = 0
theorem MeasureTheory.setAverage_sub_setAverage {α : Type u_1} {E : Type u_2} {m0 : } [] [] {μ : } {s : Set α} (hs : μ s ) (f : αE) :
∫ (x : α) in s, f x - ⨍ (a : α) in s, f aμμ = 0
theorem MeasureTheory.integral_average_sub {α : Type u_1} {E : Type u_2} {m0 : } [] [] {μ : } {f : αE} (hf : ) :
∫ (x : α), ⨍ (a : α), f aμ - f xμ = 0
theorem MeasureTheory.setIntegral_setAverage_sub {α : Type u_1} {E : Type u_2} {m0 : } [] [] {μ : } {s : Set α} {f : αE} (hs : μ s ) (hf : ) :
∫ (x : α) in s, ⨍ (a : α) in s, f aμ - f xμ = 0
theorem MeasureTheory.ofReal_average {α : Type u_1} {m0 : } {μ : } {f : α} (hf : ) (hf₀ : 0 ≤ᶠ[μ.ae] f) :
ENNReal.ofReal (⨍ (x : α), f xμ) = (∫⁻ (x : α), ENNReal.ofReal (f x)μ) / μ Set.univ
theorem MeasureTheory.ofReal_setAverage {α : Type u_1} {m0 : } {μ : } {s : Set α} {f : α} (hf : ) (hf₀ : 0 ≤ᶠ[(μ.restrict s).ae] f) :
ENNReal.ofReal (⨍ (x : α) in s, f xμ) = (∫⁻ (x : α) in s, ENNReal.ofReal (f x)μ) / μ s
theorem MeasureTheory.toReal_laverage {α : Type u_1} {m0 : } {μ : } {f : αENNReal} (hf : ) (hf' : ∀ᵐ (x : α) ∂μ, f x ) :
(⨍⁻ (x : α), f xμ).toReal = ⨍ (x : α), (f x).toRealμ
theorem MeasureTheory.toReal_setLaverage {α : Type u_1} {m0 : } {μ : } {s : Set α} {f : αENNReal} (hf : AEMeasurable f (μ.restrict s)) (hf' : ∀ᵐ (x : α) ∂μ.restrict s, f x ) :
(⨍⁻ (x : α) in s, f xμ).toReal = ⨍ (x : α) in s, (f x).toRealμ

### First moment method #

theorem MeasureTheory.measure_le_setAverage_pos {α : Type u_1} {m0 : } {μ : } {s : Set α} {f : α} (hμ : μ s 0) (hμ₁ : μ s ) (hf : ) :
0 < μ {x : α | x s f x ⨍ (a : α) in s, f aμ}

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 : } {μ : } {s : Set α} {f : α} (hμ : μ s 0) (hμ₁ : μ s ) (hf : ) :
0 < μ {x : α | x s ⨍ (a : α) in s, f aμ 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 : } {μ : } {s : Set α} {f : α} (hμ : μ s 0) (hμ₁ : μ s ) (hf : ) :
xs, f x ⨍ (a : α) in s, f aμ

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

theorem MeasureTheory.exists_setAverage_le {α : Type u_1} {m0 : } {μ : } {s : Set α} {f : α} (hμ : μ s 0) (hμ₁ : μ s ) (hf : ) :
xs, ⨍ (a : α) in s, f aμ 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 : } {μ : } {f : α} (hμ : μ 0) (hf : ) :
0 < μ {x : α | f x ⨍ (a : α), f aμ}

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 : } {μ : } {f : α} (hμ : μ 0) (hf : ) :
0 < μ {x : α | ⨍ (a : α), f aμ 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 : } {μ : } {f : α} (hμ : μ 0) (hf : ) :
∃ (x : α), f x ⨍ (a : α), f aμ

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

theorem MeasureTheory.exists_average_le {α : Type u_1} {m0 : } {μ : } {f : α} (hμ : μ 0) (hf : ) :
∃ (x : α), ⨍ (a : α), f aμ 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 : } {μ : } {N : Set α} {f : α} (hμ : μ 0) (hf : ) (hN : μ N = 0) :
xN, f x ⨍ (a : α), f aμ

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 : } {μ : } {N : Set α} {f : α} (hμ : μ 0) (hf : ) (hN : μ N = 0) :
xN, ⨍ (a : α), f aμ 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 : } {μ : } {f : α} (hf : ) :
0 < μ {x : α | f x ∫ (a : α), f aμ}

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 : } {μ : } {f : α} (hf : ) :
0 < μ {x : α | ∫ (a : α), f aμ 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 : } {μ : } {f : α} (hf : ) :
∃ (x : α), f x ∫ (a : α), f aμ

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

theorem MeasureTheory.exists_integral_le {α : Type u_1} {m0 : } {μ : } {f : α} (hf : ) :
∃ (x : α), ∫ (a : α), f aμ 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 : } {μ : } {N : Set α} {f : α} (hf : ) (hN : μ N = 0) :
xN, f x ∫ (a : α), f aμ

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 : } {μ : } {N : Set α} {f : α} (hf : ) (hN : μ N = 0) :
xN, ∫ (a : α), f aμ 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 : } {μ : } {s : Set α} {f : αENNReal} (hμ : μ s 0) (hμ₁ : μ s ) (hf : AEMeasurable f (μ.restrict s)) :
0 < μ {x : α | x s f x ⨍⁻ (a : α) in s, f aμ}

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 : } {μ : } {s : Set α} {f : αENNReal} (hμ : μ s 0) (hs : ) (hint : ∫⁻ (a : α) in s, f aμ ) :
0 < μ {x : α | x s ⨍⁻ (a : α) in s, f aμ 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 : } {μ : } {s : Set α} {f : αENNReal} (hμ : μ s 0) (hμ₁ : μ s ) (hf : AEMeasurable f (μ.restrict s)) :
xs, f x ⨍⁻ (a : α) in s, f aμ

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

theorem MeasureTheory.exists_setLaverage_le {α : Type u_1} {m0 : } {μ : } {s : Set α} {f : αENNReal} (hμ : μ s 0) (hs : ) (hint : ∫⁻ (a : α) in s, f aμ ) :
xs, ⨍⁻ (a : α) in s, f aμ 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 : } {μ : } {f : αENNReal} (hμ : μ 0) (hint : ∫⁻ (a : α), f aμ ) :
0 < μ {x : α | ⨍⁻ (a : α), f aμ 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 : } {μ : } {f : αENNReal} (hμ : μ 0) (hint : ∫⁻ (a : α), f aμ ) :
∃ (x : α), ⨍⁻ (a : α), f aμ 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 : } {μ : } {N : Set α} {f : αENNReal} (hμ : μ 0) (hint : ∫⁻ (a : α), f aμ ) (hN : μ N = 0) :
xN, ⨍⁻ (a : α), f aμ 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 : } {μ : } {f : αENNReal} (hμ : μ 0) (hf : ) :
0 < μ {x : α | f x ⨍⁻ (a : α), f aμ}

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 : } {μ : } {f : αENNReal} (hμ : μ 0) (hf : ) :
∃ (x : α), f x ⨍⁻ (a : α), f aμ

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 : } {μ : } {N : Set α} {f : αENNReal} (hμ : μ 0) (hf : ) (hN : μ N = 0) :
xN, f x ⨍⁻ (a : α), f aμ

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 : } {μ : } {f : αENNReal} (hf : ) :
0 < μ {x : α | f x ∫⁻ (a : α), f aμ}

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 : } {μ : } {f : αENNReal} (hint : ∫⁻ (a : α), f aμ ) :
0 < μ {x : α | ∫⁻ (a : α), f aμ 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 : } {μ : } {f : αENNReal} (hf : ) :
∃ (x : α), f x ∫⁻ (a : α), f aμ

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

theorem MeasureTheory.exists_lintegral_le {α : Type u_1} {m0 : } {μ : } {f : αENNReal} (hint : ∫⁻ (a : α), f aμ ) :
∃ (x : α), ∫⁻ (a : α), f aμ 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 : } {μ : } {N : Set α} {f : αENNReal} (hf : ) (hN : μ N = 0) :
xN, f x ∫⁻ (a : α), f aμ

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 : } {μ : } {N : Set α} {f : αENNReal} (hint : ∫⁻ (a : α), f aμ ) (hN : μ N = 0) :
xN, ∫⁻ (a : α), f aμ 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 : } [] [] {μ : } {ι : Type u_4} {a : ιSet α} {l : } {f : αE} {c : E} {g : ια} (K : ) (hf : Filter.Tendsto (fun (i : ι) => ⨍ (y : α) in a i, f y - cμ) l (nhds 0)) (f_int : ∀ᶠ (i : ι) in l, MeasureTheory.IntegrableOn f (a i) μ) (hg : Filter.Tendsto (fun (i : ι) => ∫ (y : α), g i yμ) 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 / (μ (a i)).toReal) :
Filter.Tendsto (fun (i : ι) => ∫ (y : α), g i y f yμ) 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.