mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
s : Set Ω
hs : MeasurableSet s
a✝ : (P.trim hm) s < ∞
⊢ Measurable P⁻[X | mΩ]

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝ : IsFiniteMeasure μ
hμν : μ ≪ ν
g : 𝓧 → 𝓨
hg : Measurable g
hσ : SigmaFinite (Measure.map g ν)
this : SigmaFinite ν
h_ne_top1 : ∀ᵐ (x : 𝓧) ∂ν, (Measure.map g μ).rnDeriv (Measure.map g ν) (g x) ≠ ∞
⊢ ∀ᵐ (x : 𝓧) ∂ν, ν⁻[μ.rnDeriv ν | MeasurableSpace.comap g m𝓨] x ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝ : IsFiniteMeasure μ
hμν : μ ≪ ν
g : 𝓧 → 𝓨
hg : Measurable g
hσ : SigmaFinite (Measure.map g ν)
this : SigmaFinite ν
h_ne_top1 : ∀ᵐ (x : 𝓧) ∂ν, (Measure.map g μ).rnDeriv (Measure.map g ν) (g x) ≠ ∞
h_ne_top2 : ∀ᵐ (x : 𝓧) ∂ν, ν⁻[μ.rnDeriv ν | MeasurableSpace.comap g m𝓨] x ≠ ∞
⊢ AEMeasurable (μ.rnDeriv ν) ν

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
φY : ↥Y → ℝ := φ ∘ ⇑Y.subtypeL
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
lem0 : ∀ᵐ (a : α) ∂μ, f a ∈ Y
lem1 : f =ᵐ[μ] fX
⊢ Integrable fY μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
φY : ↥Y → ℝ := φ ∘ ⇑Y.subtypeL
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
lem0 : ∀ᵐ (a : α) ∂μ, f a ∈ Y
lem1 : f =ᵐ[μ] fX
hfY_int : Integrable fY μ
lem2 : μ[f | m] =ᵐ[μ] ⇑Y.subtypeL ∘ μ[fY | m]
⊢ φ ∘ f =ᵐ[μ] φY ∘ fY

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
⊢ P⁻[X | mΩ] + P⁻[Y | mΩ] ≤ᵐ[P] P⁻[X + Y | mΩ]

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝ : IsFiniteMeasure μ
hμν : μ ≪ ν
g : 𝓧 → 𝓨
hg : Measurable g
hσ : SigmaFinite (Measure.map g ν)
⊢ (fun a ↦ (Measure.map g μ).rnDeriv (Measure.map g ν) (g a)) =ᵐ[ν] ν⁻[μ.rnDeriv ν | MeasurableSpace.comap g m𝓨]

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ
hf : Integrable f μ
h'f : 0 ≤ᵐ[μ] f
⊢ μ⁻[fun x ↦ ENNReal.ofReal (f x) | m] =ᵐ[μ] fun x ↦ ENNReal.ofReal (μ[f | m] x)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
hμν : μ ≪ ν
g : 𝓧 → 𝓨
hg : Measurable g
inst✝ : SigmaFinite (Measure.map g ν)
s : Set ℝ≥0∞
hs : MeasurableSet s
⊢ Measurable ((Measure.map g μ).rnDeriv (Measure.map g ν))

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
hμν : μ ≪ ν
g : 𝓧 → 𝓨
hg : Measurable g
inst✝ : SigmaFinite (Measure.map g ν)
s : Set ℝ≥0∞
hs : MeasurableSet s
⊢ g ⁻¹' (Measure.map g μ).rnDeriv (Measure.map g ν) ⁻¹' s =
    (fun a ↦ (Measure.map g μ).rnDeriv (Measure.map g ν) (g a)) ⁻¹' s

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ
hf : Integrable f μ
h'f : 0 ≤ᵐ[μ] f
x : 𝓧
hx : 0 x ≤ f x
⊢ ENNReal.ofReal (f x) = ‖f x‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
hμν : μ ≪ ν
g : 𝓧 → 𝓨
hg : Measurable g
inst✝ : SigmaFinite (Measure.map g ν)
⊢ (fun a ↦ (Measure.map g μ).rnDeriv (Measure.map g ν) (g a)) =ᵐ[ν.trim ⋯] ν⁻[μ.rnDeriv ν | MeasurableSpace.comap g m𝓨]

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ
hf : Integrable f μ
h'f : 0 ≤ᵐ[μ] f
⊢ μ⁻[fun x ↦ ENNReal.ofReal (f x) | m] =ᵐ[μ] μ⁻[fun x ↦ ‖f x‖ₑ | m]

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
φY : ↥Y → ℝ := φ ∘ ⇑Y.subtypeL
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
lem0 : ∀ᵐ (a : α) ∂μ, f a ∈ Y
lem1 : f =ᵐ[μ] fX
hfY_int : Integrable fY μ
lem2 : μ[f | m] =ᵐ[μ] ⇑Y.subtypeL ∘ μ[fY | m]
lem3 : φ ∘ f =ᵐ[μ] φY ∘ fY
⊢ φ ∘ μ[f | m] =ᵐ[μ] φY ∘ μ[fY | m]

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hX : AEMeasurable X P
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ Measurable (P⁻[X | mΩ] + P⁻[Y | mΩ])

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ
hf : Integrable f μ
h'f : 0 ≤ᵐ[μ] f
A : μ⁻[fun x ↦ ENNReal.ofReal (f x) | m] =ᵐ[μ] μ⁻[fun x ↦ ‖f x‖ₑ | m]
x : 𝓧
hx : 0 x ≤ μ[f | m] x
⊢ ENNReal.ofReal (μ[f | m] x) = ‖μ[f | m] x‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
hμν : μ ≪ ν
g : 𝓧 → 𝓨
hg : Measurable g
inst✝ : SigmaFinite (Measure.map g ν)
s : Set ℝ
hs : MeasurableSet s
⊢ Measurable fun a ↦ ((Measure.map g μ).rnDeriv (Measure.map g ν) a).toReal

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hX : AEMeasurable X P
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
s : Set Ω
hs : MeasurableSet s
⊢ Measurable P⁻[X | mΩ]

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ
hf : Integrable f μ
h'f : 0 ≤ᵐ[μ] f
⊢ μ⁻[fun x ↦ ‖f x‖ₑ | m] =ᵐ[μ] fun x ↦ ‖μ[f | m] x‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
hm : m ≤ m𝓧
μ : Measure 𝓧
inst✝ : SigmaFinite (μ.trim hm)
s : Set 𝓧
hs : MeasurableSet s
h's : μ s ≠ ∞
⊢ ∫⁻ (a : 𝓧), ‖μ[s.indicator 1 | m] a‖ₑ ∂μ = ∫⁻ (a : 𝓧), μ⁻[fun x ↦ ‖s.indicator 1 x‖ₑ | m] a ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
hm : m ≤ m𝓧
μ : Measure 𝓧
inst✝ : SigmaFinite (μ.trim hm)
s : Set 𝓧
hs : MeasurableSet s
h's : μ s ≠ ∞
⊢ ∫⁻ (a : 𝓧), μ⁻[fun x ↦ ‖s.indicator 1 x‖ₑ | m] a ∂μ = μ s

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
hμν : μ ≪ ν
g : 𝓧 → 𝓨
hg : Measurable g
inst✝ : SigmaFinite (Measure.map g ν)
⊢ (fun a ↦ ((Measure.map g μ).rnDeriv (Measure.map g ν) (g a)).toReal) =ᵐ[ν.trim ⋯]
    ν[fun a ↦ (μ.rnDeriv ν a).toReal | MeasurableSpace.comap g m𝓨]

Tactic:
by

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
g : 𝓧 → 𝓨
hμν : μ ≪ ν
hg : Measurable g
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
a : 𝓧
ha1 : ((∂Measure.map g μ/∂Measure.map g ν) (g a)).toReal = ν[fun a ↦ ((∂μ/∂ν) a).toReal | MeasurableSpace.comap g m𝓨] a
ha2 :
  f (ν[fun x ↦ ((∂μ/∂ν) x).toReal | MeasurableSpace.comap g m𝓨] a) ≤
    ν[fun x ↦ f ((∂μ/∂ν) x).toReal | MeasurableSpace.comap g m𝓨] a
⊢ f ((∂Measure.map g μ/∂Measure.map g ν) (g a)).toReal =
    f (ν[fun x ↦ ((∂μ/∂ν) x).toReal | MeasurableSpace.comap g m𝓨] a)

Tactic:
by

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
g : 𝓧 → 𝓨
hμν : μ ≪ ν
hg : Measurable g
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
⊢ (fun x ↦ f ((∂Measure.map g μ/∂Measure.map g ν) (g x)).toReal) ≤ᵐ[ν]
    ν[fun x ↦ f ((∂μ/∂ν) x).toReal | MeasurableSpace.comap g m𝓨]

Tactic:
by

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
g : 𝓧 → 𝓨
hμν : μ ≪ ν
hg : Measurable g
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ StronglyMeasurable fun x ↦ f ((∂Measure.map g μ/∂Measure.map g ν) x).toReal

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hX : AEMeasurable X P
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
s : Set Ω
hs : MeasurableSet s
⊢ AEMeasurable X (P.restrict s)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
hm : m ≤ m𝓧
inst✝ : IsFiniteMeasure μ
hsf : SigmaFinite (ν.trim hm)
hμν : μ ≪ ν
⊢ SigmaFinite (Measure.map id ν)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hX : AEMeasurable X P
⊢ P⁻[X + Y | mΩ] =ᵐ[P] P⁻[X | mΩ] + P⁻[Y | mΩ]

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hY : AEMeasurable Y P
⊢ P⁻[X + Y | mΩ] =ᵐ[P] P⁻[X | mΩ] + P⁻[Y | mΩ]

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
φY : ↥Y → ℝ := φ ∘ ⇑Y.subtypeL
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
lem0 : ∀ᵐ (a : α) ∂μ, f a ∈ Y
lem1 : f =ᵐ[μ] fX
hfY_int : Integrable fY μ
lem2 : μ[f | m] =ᵐ[μ] ⇑Y.subtypeL ∘ μ[fY | m]
lem3 : φ ∘ f =ᵐ[μ] φY ∘ fY
⊢ ContinuousOn (⇑Y.subtypeL) (⇑Y.subtypeL ⁻¹' s)

Tactic:
by

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
g : 𝓧 → 𝓨
hμν : μ ≪ ν
hg : Measurable g
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ StronglyMeasurable fun x ↦ f ((∂Measure.map g μ/∂Measure.map g ν) (g x)).toReal

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
hX : AEMeasurable X P
c : ℝ≥0∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ Measurable (c • P⁻[X | mΩ])

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
hm : m ≤ m𝓧
inst✝ : IsFiniteMeasure μ
hsf : SigmaFinite (ν.trim hm)
hμν : μ ≪ ν
⊢ (fun x ↦ ((μ.trim hm).rnDeriv (ν.trim hm) x).toReal) =ᵐ[ν.trim hm] ν[fun x ↦ (μ.rnDeriv ν x).toReal | m]

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
φY : ↥Y → ℝ := φ ∘ ⇑Y.subtypeL
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
lem0 : ∀ᵐ (a : α) ∂μ, f a ∈ Y
lem1 : f =ᵐ[μ] fX
hfY_int : Integrable fY μ
lem2 : μ[f | m] =ᵐ[μ] ⇑Y.subtypeL ∘ μ[fY | m]
lem3 : φ ∘ f =ᵐ[μ] φY ∘ fY
x : ↥Y
⊢ x ∈ ⇑Y.subtypeL ⁻¹' s → Y.subtypeL x ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
hm : m ≤ m𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : SigmaFinite (ν.trim hm)
hμν : μ ≪ ν
⊢ (μ.trim hm).rnDeriv (ν.trim hm) =ᵐ[ν.trim hm] fun x ↦ ENNReal.ofReal (ν[fun x ↦ (μ.rnDeriv ν x).toReal | m] x)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
hX : AEMeasurable X P
c : ℝ≥0∞
⊢ P⁻[c • X | mΩ] =ᵐ[P] c • P⁻[X | mΩ]

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
c : ℝ≥0∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ Measurable (c • P⁻[X | mΩ])

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
c : ℝ≥0∞
⊢ c • P⁻[X | mΩ] ≤ᵐ[P] P⁻[c • X | mΩ]

Tactic:
by

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
g : 𝓧 → 𝓨
hμν : μ ≪ ν
hg : Measurable g
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
⊢ Integrable (fun x ↦ f ((∂Measure.map g μ/∂Measure.map g ν) x).toReal) (Measure.map g ν)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
c : ℝ≥0∞
hc : c ≠ ∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ Measurable (c • P⁻[X | mΩ])

Tactic:
by

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
hm : m ≤ m𝓧
hμν : μ ≪ ν
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
a : 𝓧
ha1 : ((∂μ.trim hm/∂ν.trim hm) a).toReal = ν[fun x ↦ ((∂μ/∂ν) x).toReal | m] a
ha2 : f (ν[fun x ↦ ((∂μ/∂ν) x).toReal | m] a) ≤ ν[fun x ↦ f ((∂μ/∂ν) x).toReal | m] a
⊢ f ((∂μ.trim hm/∂ν.trim hm) a).toReal = f (ν[fun x ↦ ((∂μ/∂ν) x).toReal | m] a)

Tactic:
by

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
hm : m ≤ m𝓧
hμν : μ ≪ ν
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
⊢ (fun x ↦ f ((∂μ.trim hm/∂ν.trim hm) x).toReal) ≤ᵐ[ν.trim hm] ν[fun x ↦ f ((∂μ/∂ν) x).toReal | m]

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
c : ℝ≥0∞
hc : c ≠ ∞
⊢ P⁻[c • X | mΩ] =ᵐ[P] c • P⁻[X | mΩ]

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
φY : ↥Y → ℝ := φ ∘ ⇑Y.subtypeL
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
lem0 : ∀ᵐ (a : α) ∂μ, f a ∈ Y
lem1 : f =ᵐ[μ] fX
hfY_int : Integrable fY μ
lem2 : μ[f | m] =ᵐ[μ] ⇑Y.subtypeL ∘ μ[fY | m]
lem3 : φ ∘ f =ᵐ[μ] φY ∘ fY
⊢ φY ∘ μ[fY | m] ≤ᵐ[μ] μ[φY ∘ fY | m]

Tactic:
by

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
hm : m ≤ m𝓧
hμν : μ ≪ ν
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ StronglyMeasurable fun x ↦ f ((∂μ.trim hm/∂ν.trim hm) x).toReal

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ φ ∘ μ[f | m] ≤ᵐ[μ] μ[φ ∘ f | m]

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝ : Countable ι
X : ι → Ω → ℝ≥0∞
hX : ∀ (i : ι), AEMeasurable (X i) P
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ Measurable (∑' (i : ι), P⁻[X i | mΩ])

Tactic:
by

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
hm : m ≤ m𝓧
hμν : μ ≪ ν
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ Integrable (fun x ↦ ((∂μ.trim hm/∂ν.trim hm) x).toReal) (ν.trim hm)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝ : Countable ι
X : ι → Ω → ℝ≥0∞
hX : ∀ (i : ι), AEMeasurable (X i) P
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
s : Set Ω
hs : MeasurableSet s
⊢ ∀ (i : ι), AEMeasurable P⁻[X i | mΩ] (P.restrict s)

Tactic:
by

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
hm : m ≤ m𝓧
hμν : μ ≪ ν
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
⊢ Integrable (fun x ↦ f ((∂μ.trim hm/∂ν.trim hm) x).toReal) (ν.trim hm)

Tactic:
by

Goal state after: