# Measure theory lemmas to be upstreamed to Mathlib #

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theorem Filter.EventuallyEq.div' {α : Type u_1} {β : Type u_2} [Div β] {f f' g g' : α → β} {l : Filter α} (h : f =ᶠ[l] g) (h' : f' =ᶠ[l] g') :
f / f' =ᶠ[l] g / g'
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theorem Filter.EventuallyEq.sub' {α : Type u_1} {β : Type u_2} [Sub β] {f f' g g' : α → β} {l : Filter α} (h : f =ᶠ[l] g) (h' : f' =ᶠ[l] g') :
f - f' =ᶠ[l] g - g'
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@[simp]
theorem ProbabilityTheory.charFun_toDual_symm_eq_charFunDual {E : Type u_1} [NormedAddCommGroup E] [CompleteSpace E] [InnerProductSpace ℝ E] {mE : MeasurableSpace E} {μ : MeasureTheory.Measure E} (L : StrongDual ℝ E) :
MeasureTheory.charFun μ ((InnerProductSpace.toDual ℝ E).symm L) = MeasureTheory.charFunDual μ L
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theorem ProbabilityTheory.eq_gaussianReal_integral_variance {μ : MeasureTheory.Measure ℝ} {m : ℝ} {v : NNReal} (h : μ = gaussianReal m v) :
μ = gaussianReal (∫ (x : ℝ), id x ∂μ) (variance id μ).toNNReal
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theorem ProbabilityTheory.variance_pi {ι : Type u_1} [Fintype ι] {Ω : ι → Type u_2} {mΩ : (i : ι) → MeasurableSpace (Ω i)} {μ : (i : ι) → MeasureTheory.Measure (Ω i)} [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] {X : (i : ι) → Ω i → ℝ} (h : ∀ (i : ι), MeasureTheory.MemLp (X i) 2 (μ i)) :
variance (∑ i : ι, fun (ω : (i : ι) → Ω i) => X i (ω i)) (MeasureTheory.Measure.pi μ) = ∑ i : ι, variance (X i) (μ i)
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theorem ProbabilityTheory.centralMoment_of_integral_id_eq_zero {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} (p : ℕ) (hX : ∫ (x : Ω), X x ∂μ = 0) :
centralMoment X p μ = ∫ (ω : Ω), X ω ^ p ∂μ
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theorem MeasurableEquiv.coe_toLp_symm_eq {ι : Type u_1} :
⇑(MeasurableEquiv.toLp 2 (ι → ℝ)).symm = ⇑(EuclideanSpace.equiv ι ℝ)
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@[simp]
theorem zero_mem_parallelepiped {ι : Type u_1} {E : Type u_2} [Fintype ι] [AddCommGroup E] [Module ℝ E] {v : ι → E} :
0 ∈ parallelepiped v
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@[simp]
theorem nonempty_parallelepiped {ι : Type u_1} {E : Type u_2} [Fintype ι] [AddCommGroup E] [Module ℝ E] {v : ι → E} :
(parallelepiped v).Nonempty
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@[simp]
theorem volume_of_nonempty_of_subsingleton {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [Subsingleton E] {s : Set E} (hs : s.Nonempty) :
MeasureTheory.volume s = 1
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theorem MeasureTheory.Measure.IsMulLeftInvariant.measure_ball_const {G : Type u_1} [Group G] [PseudoMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsMulLeftInvariant] [IsIsometricSMul G G] [MeasurableMul G] (a b : G) (r : ℝ) :
μ (Metric.ball a r) = μ (Metric.ball b r)
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theorem MeasureTheory.Measure.IsAddLeftInvariant.measure_ball_const {G : Type u_1} [AddGroup G] [PseudoMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsAddLeftInvariant] [IsIsometricVAdd G G] [MeasurableAdd G] (a b : G) (r : ℝ) :
μ (Metric.ball a r) = μ (Metric.ball b r)
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theorem MeasureTheory.Measure.IsMulRightInvariant.measure_ball_const {G : Type u_1} [CommGroup G] [PseudoMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsMulRightInvariant] [IsIsometricSMul G ᵐᵒᵖ G] [MeasurableMul G] (a b : G) (r : ℝ) :
μ (Metric.ball a r) = μ (Metric.ball b r)
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theorem MeasureTheory.Measure.IsAddRightInvariant.measure_ball_const {G : Type u_1} [AddCommGroup G] [PseudoMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsAddRightInvariant] [IsIsometricVAdd G ᵃᵒᵖ G] [MeasurableAdd G] (a b : G) (r : ℝ) :
μ (Metric.ball a r) = μ (Metric.ball b r)
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theorem MeasureTheory.Measure.IsMulLeftInvariant.measure_closedBall_const {G : Type u_1} [Group G] [PseudoMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsMulLeftInvariant] [IsIsometricSMul G G] [MeasurableMul G] (a b : G) (r : ℝ) :
μ (Metric.closedBall a r) = μ (Metric.closedBall b r)
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theorem MeasureTheory.Measure.IsAddLeftInvariant.measure_closedBall_const {G : Type u_1} [AddGroup G] [PseudoMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsAddLeftInvariant] [IsIsometricVAdd G G] [MeasurableAdd G] (a b : G) (r : ℝ) :
μ (Metric.closedBall a r) = μ (Metric.closedBall b r)
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theorem MeasureTheory.Measure.IsMulRightInvariant.measure_closeBall_const {G : Type u_1} [CommGroup G] [PseudoMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsMulRightInvariant] [IsIsometricSMul G ᵐᵒᵖ G] [MeasurableMul G] (a b : G) (r : ℝ) :
μ (Metric.closedBall a r) = μ (Metric.closedBall b r)
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theorem MeasureTheory.Measure.IsAddRightInvariant.measure_closeBall_const {G : Type u_1} [AddCommGroup G] [PseudoMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsAddRightInvariant] [IsIsometricVAdd G ᵃᵒᵖ G] [MeasurableAdd G] (a b : G) (r : ℝ) :
μ (Metric.closedBall a r) = μ (Metric.closedBall b r)
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theorem MeasureTheory.Measure.IsMulLeftInvariant.measure_ball_const' {G : Type u_1} [Group G] [PseudoEMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsMulLeftInvariant] [IsIsometricSMul G G] [MeasurableMul G] (a b : G) (r : ENNReal) :
μ (EMetric.ball a r) = μ (EMetric.ball b r)
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theorem MeasureTheory.Measure.IsAddLeftInvariant.measure_ball_const' {G : Type u_1} [AddGroup G] [PseudoEMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsAddLeftInvariant] [IsIsometricVAdd G G] [MeasurableAdd G] (a b : G) (r : ENNReal) :
μ (EMetric.ball a r) = μ (EMetric.ball b r)
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theorem MeasureTheory.Measure.IsMulRightInvariant.measure_ball_const' {G : Type u_1} [CommGroup G] [PseudoEMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsMulRightInvariant] [IsIsometricSMul G ᵐᵒᵖ G] [MeasurableMul G] (a b : G) (r : ENNReal) :
μ (EMetric.ball a r) = μ (EMetric.ball b r)
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theorem MeasureTheory.Measure.IsAddRightInvariant.measure_ball_const' {G : Type u_1} [AddCommGroup G] [PseudoEMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsAddRightInvariant] [IsIsometricVAdd G ᵃᵒᵖ G] [MeasurableAdd G] (a b : G) (r : ENNReal) :
μ (EMetric.ball a r) = μ (EMetric.ball b r)
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theorem MeasureTheory.Measure.IsMulLeftInvariant.measure_closedBall_const' {G : Type u_1} [Group G] [PseudoEMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsMulLeftInvariant] [IsIsometricSMul G G] [MeasurableMul G] (a b : G) (r : ENNReal) :
μ (EMetric.closedBall a r) = μ (EMetric.closedBall b r)
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theorem MeasureTheory.Measure.IsAddLeftInvariant.measure_closedBall_const' {G : Type u_1} [AddGroup G] [PseudoEMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsAddLeftInvariant] [IsIsometricVAdd G G] [MeasurableAdd G] (a b : G) (r : ENNReal) :
μ (EMetric.closedBall a r) = μ (EMetric.closedBall b r)
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theorem MeasureTheory.Measure.IsMulRightInvariant.measure_closeBall_const' {G : Type u_1} [CommGroup G] [PseudoEMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsMulRightInvariant] [IsIsometricSMul G ᵐᵒᵖ G] [MeasurableMul G] (a b : G) (r : ENNReal) :
μ (EMetric.closedBall a r) = μ (EMetric.closedBall b r)
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theorem MeasureTheory.Measure.IsAddRightInvariant.measure_closeBall_const' {G : Type u_1} [AddCommGroup G] [PseudoEMetricSpace G] [MeasurableSpace G] [OpensMeasurableSpace G] (μ : Measure G) [μ.IsAddRightInvariant] [IsIsometricVAdd G ᵃᵒᵖ G] [MeasurableAdd G] (a b : G) (r : ENNReal) :
μ (EMetric.closedBall a r) = μ (EMetric.closedBall b r)
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theorem InnerProductSpace.volume_closedBall_div {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (x y : E) {r s : ℝ} (hr : 0 < r) (hs : 0 < s) :
MeasureTheory.volume (Metric.closedBall x r) / MeasureTheory.volume (Metric.closedBall y s) = ENNReal.ofReal (r / s) ^ Module.finrank ℝ E
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theorem InnerProductSpace.volume_closedBall_div' {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (x y : E) (r s : ENNReal) :
MeasureTheory.volume (EMetric.closedBall x r) / MeasureTheory.volume (EMetric.closedBall y s) = (r / s) ^ Module.finrank ℝ E
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theorem ProbabilityTheory.covariance_fun_add_left {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X Y Z : Ω → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hX : MeasureTheory.MemLp X 2 μ) (hY : MeasureTheory.MemLp Y 2 μ) (hZ : MeasureTheory.MemLp Z 2 μ) :
covariance (fun (ω : Ω) => X ω + Y ω) Z μ = covariance (fun (ω : Ω) => X ω) Z μ + covariance (fun (ω : Ω) => Y ω) Z μ
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theorem ProbabilityTheory.covariance_fun_add_right {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X Y Z : Ω → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hX : MeasureTheory.MemLp X 2 μ) (hY : MeasureTheory.MemLp Y 2 μ) (hZ : MeasureTheory.MemLp Z 2 μ) :
covariance X (fun (ω : Ω) => Y ω + Z ω) μ = covariance X (fun (ω : Ω) => Y ω) μ + covariance X (fun (ω : Ω) => Z ω) μ
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theorem ProbabilityTheory.covariance_fun_sub_left {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X Y Z : Ω → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hX : MeasureTheory.MemLp X 2 μ) (hY : MeasureTheory.MemLp Y 2 μ) (hZ : MeasureTheory.MemLp Z 2 μ) :
covariance (fun (ω : Ω) => X ω - Y ω) Z μ = covariance X Z μ - covariance Y Z μ
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theorem ProbabilityTheory.covariance_fun_sub_right {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X Y Z : Ω → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hX : MeasureTheory.MemLp X 2 μ) (hY : MeasureTheory.MemLp Y 2 μ) (hZ : MeasureTheory.MemLp Z 2 μ) :
covariance X (fun (ω : Ω) => Y ω - Z ω) μ = covariance X (fun (ω : Ω) => Y ω) μ - covariance X (fun (ω : Ω) => Z ω) μ
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theorem ProbabilityTheory.covariance_fun_div_left {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X Y : Ω → ℝ} (c : ℝ) :
covariance (fun (ω : Ω) => X ω / c) Y μ = covariance X Y μ / c
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theorem ProbabilityTheory.covariance_fun_div_right {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X Y : Ω → ℝ} (c : ℝ) :
covariance X (fun (ω : Ω) => Y ω / c) μ = covariance X Y μ / c
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theorem ProbabilityTheory.variance_fun_div {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} (c : ℝ) (hX : AEMeasurable X μ) :
variance (fun (ω : Ω) => X ω / c) μ = variance X μ / c ^ 2
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theorem MeasurableSpace.comap_process {Ω : Type u_1} {T : Type u_2} {𝓧 : T → Type u_3} [(t : T) → MeasurableSpace (𝓧 t)] (X : (t : T) → Ω → 𝓧 t) :
MeasurableSpace.comap (fun (ω : Ω) (t : T) => X t ω) pi = ⨆ (t : T), MeasurableSpace.comap (X t) inferInstance
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theorem MeasurableSpace.comap_le_comap {Ω : Type u_1} {𝓧 : Type u_2} {𝓨 : Type u_3} [m𝓧 : MeasurableSpace 𝓧] [m𝓨 : MeasurableSpace 𝓨] {X : Ω → 𝓧} {Y : Ω → 𝓨} (f : 𝓧 → 𝓨) (hf : Measurable f) (h : Y = f ∘ X) :
MeasurableSpace.comap Y m𝓨 ≤ MeasurableSpace.comap X m𝓧
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theorem MeasureTheory.Filtration.natural_eq_comap {Ω : Type u_1} {ι : Type u_2} {m : MeasurableSpace Ω} {β : ι → Type u_3} [(i : ι) → TopologicalSpace (β i)] [∀ (i : ι), TopologicalSpace.MetrizableSpace (β i)] [mβ : (i : ι) → MeasurableSpace (β i)] [∀ (i : ι), BorelSpace (β i)] [Preorder ι] (u : (i : ι) → Ω → β i) (hum : ∀ (i : ι), StronglyMeasurable (u i)) (i : ι) :
↑(natural u hum) i = MeasurableSpace.comap (fun (ω : Ω) (j : ↑(Set.Iic i)) => u (↑j) ω) inferInstance
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theorem ProbabilityTheory.measure_eq_zero_or_one_of_indep_self {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω} [MeasureTheory.IsZeroOrProbabilityMeasure P] (hm1 : m ≤ mΩ) (hm2 : Indep m m P) {A : Set Ω} (hA : MeasurableSet A) :
P A = 0 ∨ P A = 1
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theorem MeasurableSpace.generateFrom_singleton_eq_comap_indicator_one {Ω : Type u_1} {A : Set Ω} :
generateFrom {A} = MeasurableSpace.comap (A.indicator 1) inferInstance
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theorem ProbabilityTheory.singleton_indepSets_comap_iff {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω} [MeasureTheory.IsZeroOrProbabilityMeasure P] {𝓧 : Type u_2} {m𝓧 : MeasurableSpace 𝓧} {A : Set Ω} {X : Ω → 𝓧} (hX : Measurable X) (hA : MeasurableSet A) :
IndepSets {A} {s : Set Ω | MeasurableSet s} P ↔ IndepFun (A.indicator 1) X P
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theorem IndepSets.setIntegral_eq_mul {Ω : Type u_1} {𝓧 : Type u_2} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {m𝓧 : MeasurableSpace 𝓧} {X : Ω → 𝓧} [MeasureTheory.IsZeroOrProbabilityMeasure μ] {f : 𝓧 → ℝ} {A : Set Ω} (hA1 : ProbabilityTheory.IndepSets {A} {s : Set Ω | MeasurableSet s} μ) (hX : Measurable X) (hA2 : MeasurableSet A) (hf : MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map X μ)) :
∫ (ω : Ω) in A, f (X ω) ∂μ = μ.real A * ∫ (ω : Ω), f (X ω) ∂μ
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theorem Indep.singleton_indepSets {Ω : Type u_1} {m1 m2 mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω} (h : ProbabilityTheory.Indep m1 m2 P) {A : Set Ω} (hA : MeasurableSet A) :
ProbabilityTheory.IndepSets {A} {s : Set Ω | MeasurableSet s} P
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theorem measurableSpace_le_iff {Ω : Type u_1} {m1 m2 : MeasurableSpace Ω} :
m1 ≤ m2 ↔ ∀ (s : Set Ω), MeasurableSet s → MeasurableSet s