Measure theory lemmas to be upstreamed to Mathlib

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'

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'

theorem MeasureTheory.MemLp.aemeasurable {X : Type u_1} {Y : Type u_2} {mX : MeasurableSpace X} {μ : Measure X} [MeasurableSpace Y] [ENorm Y] [TopologicalSpace Y] [TopologicalSpace.PseudoMetrizableSpace Y] [BorelSpace Y] {f : X → Y} {p : ENNReal} (hf : MemLp f p μ) : AEMeasurable f μ

theorem ProbabilityTheory.charFun_pi {ι : Type u_1} [Fintype ι] {E : ι → Type u_2} {mE : (i : ι) → MeasurableSpace (E i)} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → InnerProductSpace ℝ (E i)] (μ : (i : ι) → MeasureTheory.Measure (E i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (t : PiLp 2 E) : MeasureTheory.charFun (MeasureTheory.Measure.pi μ) t = ∏ i : ι, MeasureTheory.charFun (μ i) (t i)

@[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 : NormedSpace.Dual ℝ E) : MeasureTheory.charFun μ ((InnerProductSpace.toDual ℝ E).symm L) = MeasureTheory.charFunDual μ L

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

theorem ProbabilityTheory.measurePreserving_eval {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)] (i : ι) : MeasureTheory.MeasurePreserving (Function.eval i) (MeasureTheory.Measure.pi μ) (μ i)

theorem ProbabilityTheory.iIndepFun_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 : ι → Type u_3} {m𝒳 : (i : ι) → MeasurableSpace (𝒳 i)} {X : (i : ι) → Ω i → 𝒳 i} (mX : ∀ (i : ι), Measurable (X i)) : iIndepFun (fun (i : ι) (ω : (i : ι) → Ω i) => X i (ω i)) (MeasureTheory.Measure.pi μ)

theorem ProbabilityTheory.iIndepFun_pi₀ {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 : ι → Type u_3} {m𝒳 : (i : ι) → MeasurableSpace (𝒳 i)} {X : (i : ι) → Ω i → 𝒳 i} (mX : ∀ (i : ι), AEMeasurable (X i) (μ i)) : iIndepFun (fun (i : ι) (ω : (i : ι) → Ω i) => X i (ω i)) (MeasureTheory.Measure.pi μ)

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)

theorem ProbabilityTheory.variance_sub {Ω : Type u_4} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {X Y : Ω → ℝ} (hX : MeasureTheory.MemLp X 2 μ) (hY : MeasureTheory.MemLp Y 2 μ) : variance (X - Y) μ = variance X μ - 2 * covariance X Y μ + variance Y μ

theorem ProbabilityTheory.covariance_map_equiv {Ω : Type u_4} {Ω' : Type u_5} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} {μ : MeasureTheory.Measure Ω'} (X Y : Ω → ℝ) (Z : Ω' ≃ᵐ Ω) : covariance X Y (MeasureTheory.Measure.map (⇑Z) μ) = covariance (X ∘ ⇑Z) (Y ∘ ⇑Z) μ

theorem ProbabilityTheory.variance_map_equiv {Ω : Type u_4} {Ω' : Type u_5} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} {μ : MeasureTheory.Measure Ω'} (X : Ω → ℝ) (Y : Ω' ≃ᵐ Ω) : variance X (MeasureTheory.Measure.map (⇑Y) μ) = variance (X ∘ ⇑Y) μ

theorem ProbabilityTheory.centralMoment_of_integral_id_eq_zero {Ω : Type u_4} {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

theorem ContinuousLinearMap.integral_comp_id_comm' {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace ℝ E] [NormedSpace 𝕜 F] [NormedSpace ℝ F] [CompleteSpace E] [CompleteSpace F] [MeasurableSpace E] {μ : MeasureTheory.Measure E} (h : MeasureTheory.Integrable _root_.id μ) (L : E →L[𝕜] F) : ∫ (x : E), L x ∂μ = L (∫ (x : E), _root_.id x ∂μ)

theorem ContinuousLinearMap.integral_comp_id_comm {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace ℝ E] [NormedSpace 𝕜 F] [NormedSpace ℝ F] [CompleteSpace E] [CompleteSpace F] [MeasurableSpace E] {μ : MeasureTheory.Measure E} (h : MeasureTheory.Integrable _root_.id μ) (L : E →L[𝕜] F) : ∫ (x : E), L x ∂μ = L (∫ (x : E), x ∂μ)

theorem ContinuousLinearMap.integral_id_map {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace ℝ E] [NormedSpace 𝕜 F] [NormedSpace ℝ F] [CompleteSpace E] [CompleteSpace F] [MeasurableSpace E] {μ : MeasureTheory.Measure E} [OpensMeasurableSpace E] [MeasurableSpace F] [BorelSpace F] [SecondCountableTopology F] (h : MeasureTheory.Integrable _root_.id μ) (L : E →L[𝕜] F) : ∫ (x : F), _root_.id x ∂MeasureTheory.Measure.map (⇑L) μ = L (∫ (x : E), _root_.id x ∂μ)

theorem EuclideanSpace.coe_measurableEquiv' {ι : Type u_1} : ⇑(EuclideanSpace.measurableEquiv ι) = ⇑(equiv ι ℝ)