mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.InformationTheory.KullbackLeibler.ChainRule

Goal state before:
pp: case neg
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ : Kernel 𝓧 𝓨
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsMarkovKernel κ
h_ac : ¬μ ⊗ₘ κ ≪ ν ⊗ₘ κ
⊢ klDiv (μ ⊗ₘ κ) (ν ⊗ₘ κ) = klDiv μ ν

Tactic:
rw [klDiv_of_not_ac h_ac, klDiv_of_not_ac]

Goal state after:
case neg
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ : Kernel 𝓧 𝓨
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsMarkovKernel κ
h_ac : ¬μ ⊗ₘ κ ≪ ν ⊗ₘ κ
⊢ ¬μ ≪ ν
Module: Mathlib.InformationTheory.KullbackLeibler.ChainRule

Goal state before:
pp: case neg
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ : Kernel 𝓧 𝓨
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsMarkovKernel κ
h_ac : ¬μ ≪ ν
⊢ ¬μ ≪ ν

Tactic:
]

Goal state after:
case neg
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ : Kernel 𝓧 𝓨
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsMarkovKernel κ
h_ac : ¬μ ≪ ν
⊢ ¬μ ≪ ν
Module: Mathlib.InformationTheory.KullbackLeibler.ChainRule

Goal state before:
pp: case pos
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ : Kernel 𝓧 𝓨
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsMarkovKernel κ
h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ κ
⊢ ∫⁻ (x : 𝓧 × 𝓨), ENNReal.ofReal (klFun ((∂μ ⊗ₘ κ/∂ν ⊗ₘ κ) x).toReal) ∂ν ⊗ₘ κ =
    ∫⁻ (x : 𝓧), ENNReal.ofReal (klFun ((∂μ/∂ν) x).toReal) ∂ν

Tactic:
]

Goal state after:
case pos
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ : Kernel 𝓧 𝓨
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsMarkovKernel κ
h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ κ
⊢ ∫⁻ (x : 𝓧 × 𝓨), ENNReal.ofReal (klFun ((∂μ ⊗ₘ κ/∂ν ⊗ₘ κ) x).toReal) ∂ν ⊗ₘ κ =
    ∫⁻ (x : 𝓧), ENNReal.ofReal (klFun ((∂μ/∂ν) x).toReal) ∂ν
Module: Mathlib.InformationTheory.KullbackLeibler.ChainRule

Goal state before:
pp: case pos
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ : Kernel 𝓧 𝓨
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsMarkovKernel κ
h_ac : μ ≪ ν
⊢ ∫⁻ (x : 𝓧 × 𝓨), ENNReal.ofReal (klFun ((∂μ ⊗ₘ κ/∂ν ⊗ₘ κ) x).toReal) ∂ν ⊗ₘ κ =
    ∫⁻ (x : 𝓧), ENNReal.ofReal (klFun ((∂μ/∂ν) x).toReal) ∂ν

Tactic:
]

Goal state after:
case pos
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ : Kernel 𝓧 𝓨
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsMarkovKernel κ
h_ac : μ ≪ ν
⊢ ∫⁻ (x : 𝓧 × 𝓨), ENNReal.ofReal (klFun ((∂μ ⊗ₘ κ/∂ν ⊗ₘ κ) x).toReal) ∂ν ⊗ₘ κ =
    ∫⁻ (x : 𝓧), ENNReal.ofReal (klFun ((∂μ/∂ν) x).toReal) ∂ν
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case pos
α : Type u_1
E : Type u_3
f : α → E
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
m₁ m₂ m₀ : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m₀
inst✝ : SigmaFinite (μ.trim hm₂)
hμm₁ : SigmaFinite (μ.trim ⋯)
hf : Integrable f μ
s : Set α
hs : MeasurableSet s
a✝ : μ s < ∞
⊢ ∫ (x : α) in s, μ[f | m₂] x ∂μ = ∫ (x : α) in s, μ[f | m₁] x ∂μ

Tactic:
]

Goal state after:
case pos
α : Type u_1
E : Type u_3
f : α → E
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
m₁ m₂ m₀ : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m₀
inst✝ : SigmaFinite (μ.trim hm₂)
hμm₁ : SigmaFinite (μ.trim ⋯)
hf : Integrable f μ
s : Set α
hs : MeasurableSet s
a✝ : μ s < ∞
⊢ ∫ (x : α) in s, μ[f | m₂] x ∂μ = ∫ (x : α) in s, μ[f | m₁] x ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case pos
α : Type u_1
E : Type u_3
f : α → E
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
m₁ m₂ m₀ : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m₀
inst✝ : SigmaFinite (μ.trim hm₂)
hμm₁ : SigmaFinite (μ.trim ⋯)
hf : Integrable f μ
s : Set α
hs : MeasurableSet s
a✝ : μ s < ∞
⊢ ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, f x ∂μ

Tactic:
]

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

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ : Kernel 𝓧 𝓨
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsMarkovKernel κ
h_ac : μ ≪ ν
⊢ ∫⁻ (a : 𝓧), ∫⁻ (b : 𝓨), ENNReal.ofReal (klFun ((∂μ/∂ν) (a, b).1).toReal) ∂κ a ∂ν =
    ∫⁻ (x : 𝓧), ENNReal.ofReal (klFun ((∂μ/∂ν) x).toReal) ∂ν

Tactic:
]

Goal state after:
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ : Kernel 𝓧 𝓨
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsMarkovKernel κ
h_ac : μ ≪ ν
⊢ ∫⁻ (a : 𝓧), ∫⁻ (b : 𝓨), ENNReal.ofReal (klFun ((∂μ/∂ν) (a, b).1).toReal) ∂κ a ∂ν =
    ∫⁻ (x : 𝓧), ENNReal.ofReal (klFun ((∂μ/∂ν) x).toReal) ∂ν
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hf_int : Integrable f μ
hs : MeasurableSet s
hm : m ≤ m0
hμm this : SigmaFinite (μ.trim hm)
⊢ s.indicator μ[f | m] =ᵐ[μ] s.indicator μ[f | m]

Tactic:
]

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

Goal state before:
pp: case neg
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ η : Kernel 𝓧 𝓨
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
h_ac : ¬μ ⊗ₘ κ ≪ ν ⊗ₘ η
⊢ ∞ = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)

Tactic:
]

Goal state after:
case neg
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ η : Kernel 𝓧 𝓨
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
h_ac : ¬μ ⊗ₘ κ ≪ ν ⊗ₘ η
⊢ ∞ = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)
Module: Mathlib.InformationTheory.KullbackLeibler.ChainRule

Goal state before:
pp: case neg
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ η : Kernel 𝓧 𝓨
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
h_ac : ¬(μ ≪ ν ∧ μ ⊗ₘ κ ≪ μ ⊗ₘ η)
⊢ ∞ = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)

Tactic:
]

Goal state after:
case neg
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ η : Kernel 𝓧 𝓨
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
h_ac : ¬(μ ≪ ν ∧ μ ⊗ₘ κ ≪ μ ⊗ₘ η)
⊢ ∞ = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hf_int : Integrable f μ
hs : MeasurableSet s
hm : m ≤ m0
hμm this✝ : SigmaFinite (μ.trim hm)
this : s.indicator μ[f | m] =ᵐ[μ] s.indicator μ[s.indicator f + sᶜ.indicator f | m]
x : α
hx : μ[s.indicator f + sᶜ.indicator f | m] x = (μ[s.indicator f | m] + μ[sᶜ.indicator f | m]) x
⊢ (if x ∈ s then (μ[s.indicator f | m] + μ[sᶜ.indicator f | m]) x else 0) =
    if x ∈ s then (μ[s.indicator f | m] + μ[sᶜ.indicator f | m]) x else 0

Tactic:
]

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

Goal state before:
pp: case neg
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ η : Kernel 𝓧 𝓨
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η
h_int : ¬Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ)
⊢ ∞ = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)

Tactic:
]

Goal state after:
case neg
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ η : Kernel 𝓧 𝓨
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η
h_int : ¬Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ)
⊢ ∞ = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)
Module: Mathlib.InformationTheory.KullbackLeibler.ChainRule

Goal state before:
pp: case neg
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ η : Kernel 𝓧 𝓨
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η
h_int : ¬(Integrable (llr μ ν) μ ∧ Integrable (llr (μ ⊗ₘ κ) (μ ⊗ₘ η)) (μ ⊗ₘ κ))
⊢ ∞ = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)

Tactic:
]

Goal state after:
case neg
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ η : Kernel 𝓧 𝓨
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η
h_int : ¬(Integrable (llr μ ν) μ ∧ Integrable (llr (μ ⊗ₘ κ) (μ ⊗ₘ η)) (μ ⊗ₘ κ))
⊢ ∞ = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: case refine_2
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hf_int : Integrable f μ
hs : MeasurableSet s
hm : m ≤ m0
hμm this✝ : SigmaFinite (μ.trim hm)
this : s.indicator μ[f | m] =ᵐ[μ] s.indicator μ[s.indicator f + sᶜ.indicator f | m]
⊢ μ[sᶜ.indicator f | m] =ᵐ[μ] μ[sᶜ.indicator f | m]

Tactic:
]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hf_int : Integrable f μ
hs : MeasurableSet s
hm : m ≤ m0
hμm this✝ : SigmaFinite (μ.trim hm)
this : s.indicator μ[f | m] =ᵐ[μ] s.indicator μ[s.indicator f + sᶜ.indicator f | m]
⊢ s.indicator μ[s.indicator f | m] =ᵐ[μ] s.indicator μ[s.indicator f | m]

Tactic:
]

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

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ η : Kernel 𝓧 𝓨
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η
h_int : Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ)
h_ac_μν : μ ≪ ν
h_ac_κη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
h_int_μν : Integrable (llr μ ν) μ
h_int_κη : Integrable (llr (μ ⊗ₘ κ) (μ ⊗ₘ η)) (μ ⊗ₘ κ)
this :
  ∫ (p : 𝓧 × 𝓨), llr (μ ⊗ₘ κ) (ν ⊗ₘ η) p ∂μ ⊗ₘ κ =
    ∫ (a : 𝓧), llr μ ν a ∂μ + ∫ (p : 𝓧 × 𝓨), llr (μ ⊗ₘ κ) (μ ⊗ₘ η) p ∂μ ⊗ₘ κ
⊢ ENNReal.ofReal
      (∫ (a : 𝓧), llr μ ν a ∂μ + ∫ (p : 𝓧 × 𝓨), llr (μ ⊗ₘ κ) (μ ⊗ₘ η) p ∂μ ⊗ₘ κ + (ν univ).toReal - (μ univ).toReal) =
    ENNReal.ofReal
      (∫ (x : 𝓧), llr μ ν x ∂μ + (ν univ).toReal - (μ univ).toReal + ∫ (x : 𝓧 × 𝓨), llr (μ ⊗ₘ κ) (μ ⊗ₘ η) x ∂μ ⊗ₘ κ)

Tactic:
]

Goal state after:
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ η : Kernel 𝓧 𝓨
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η
h_int : Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ)
h_ac_μν : μ ≪ ν
h_ac_κη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
h_int_μν : Integrable (llr μ ν) μ
h_int_κη : Integrable (llr (μ ⊗ₘ κ) (μ ⊗ₘ η)) (μ ⊗ₘ κ)
this :
  ∫ (p : 𝓧 × 𝓨), llr (μ ⊗ₘ κ) (ν ⊗ₘ η) p ∂μ ⊗ₘ κ =
    ∫ (a : 𝓧), llr μ ν a ∂μ + ∫ (p : 𝓧 × 𝓨), llr (μ ⊗ₘ κ) (μ ⊗ₘ η) p ∂μ ⊗ₘ κ
⊢ ENNReal.ofReal
      (∫ (a : 𝓧), llr μ ν a ∂μ + ∫ (p : 𝓧 × 𝓨), llr (μ ⊗ₘ κ) (μ ⊗ₘ η) p ∂μ ⊗ₘ κ + (ν univ).toReal - (μ univ).toReal) =
    ENNReal.ofReal
      (∫ (x : 𝓧), llr μ ν x ∂μ + (ν univ).toReal - (μ univ).toReal + ∫ (x : 𝓧 × 𝓨), llr (μ ⊗ₘ κ) (μ ⊗ₘ η) x ∂μ ⊗ₘ κ)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: case refine_2
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hf_int : Integrable f μ
hs : MeasurableSet s
hm : m ≤ m0
hμm this✝ : SigmaFinite (μ.trim hm)
this : s.indicator μ[f | m] =ᵐ[μ] s.indicator μ[s.indicator f + sᶜ.indicator f | m]
⊢ μ[s.indicator f | m] =ᵐ[μ] μ[s.indicator f | m]

Tactic:
]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
⊢ SigmaFinite ((μ.trim hm).restrict s)

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
⊢ SigmaFinite ((μ.trim hm).restrict s)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: α : Type u_1
E : Type u_3
𝕜 : Type u_4
inst✝⁵ : RCLike 𝕜
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : InnerProductSpace 𝕜 E
hm : m ≤ m₀
hf1 : Integrable f μ
hf2 : MemLp f 2 μ
inst✝ : SigmaFinite (μ.trim hm)
s : Set α
hs : MeasurableSet s
htop : μ s < ∞
⊢ ∫ (x : α) in s, ↑↑(toLp f hf2) x ∂μ = ∫ (x : α) in s, f x ∂μ

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_3
𝕜 : Type u_4
inst✝⁵ : RCLike 𝕜
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : InnerProductSpace 𝕜 E
hm : m ≤ m₀
hf1 : Integrable f μ
hf2 : MemLp f 2 μ
inst✝ : SigmaFinite (μ.trim hm)
s : Set α
hs : MeasurableSet s
htop : μ s < ∞
⊢ ∫ (x : α) in s, ↑↑(toLp f hf2) x ∂μ = ∫ (x : α) in s, f x ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)
⊢ s.indicator (μ.restrict s)[f | m] =ᵐ[μ] s.indicator μ[f | m]

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)
⊢ s.indicator (μ.restrict s)[f | m] =ᵐ[μ] s.indicator μ[f | m]
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: case neg
α : Type u_1
F' : Type u_3
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : α → F'
hF' : CompleteSpace ↥(Lp F' 1 μ)
hf : ¬Integrable f μ
⊢ AEStronglyMeasurable (↑↑0) μ

Tactic:
]

Goal state after:
case neg
α : Type u_1
F' : Type u_3
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : α → F'
hF' : CompleteSpace ↥(Lp F' 1 μ)
hf : ¬Integrable f μ
⊢ AEStronglyMeasurable (↑↑0) μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: case refine_1
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)
t : Set α
ht : MeasurableSet t
a✝ : μ t < ∞
⊢ Integrable (s.indicator (t.indicator (μ.restrict s)[f | m])) μ

Tactic:
]

Goal state after:
case refine_1
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)
t : Set α
ht : MeasurableSet t
a✝ : μ t < ∞
⊢ Integrable (s.indicator (t.indicator (μ.restrict s)[f | m])) μ
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
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hf_int : Integrable f μ
hs : IsClosed s
hc : Convex ℝ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
⊢ ∀ᵐ (a : α) ∂μ, μ[f | m] a ∈ s

Tactic:
borelize E

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hf_int : Integrable f μ
hs : IsClosed s
hc : Convex ℝ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ ∀ᵐ (a : α) ∂μ, μ[f | m] a ∈ s
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: α : Type u_1
F' : Type u_3
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : α → F'
hF' : CompleteSpace ↥(Lp F' 1 μ)
hf : Integrable f μ
hf' : ¬f =ᵐ[μ] 0
⊢ ¬f =ᵐ[μ] 0

Tactic:
]

Goal state after:
α : Type u_1
F' : Type u_3
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : α → F'
hF' : CompleteSpace ↥(Lp F' 1 μ)
hf : Integrable f μ
hf' : ¬f =ᵐ[μ] 0
⊢ ¬f =ᵐ[μ] 0
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)
t : Set α
ht : MeasurableSet t
a✝ : μ t < ∞
h_int_restrict : Integrable (t.indicator (μ.restrict s)[f | m]) (μ.restrict s)
⊢ Integrable (t.indicator (μ.restrict s)[f | m]) (μ.restrict s)

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)
t : Set α
ht : MeasurableSet t
a✝ : μ t < ∞
h_int_restrict : Integrable (t.indicator (μ.restrict s)[f | m]) (μ.restrict s)
⊢ Integrable (t.indicator (μ.restrict s)[f | m]) (μ.restrict s)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: case neg
α : Type u_1
F' : Type u_3
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : α → F'
hF' : CompleteSpace ↥(Lp F' 1 μ)
hf : Integrable f μ
hf' : ¬f =ᵐ[μ] 0
this : CompleteSpace F'
⊢ AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) (Integrable.toL1 f hf))) μ

Tactic:
]

Goal state after:
case neg
α : Type u_1
F' : Type u_3
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : α → F'
hF' : CompleteSpace ↥(Lp F' 1 μ)
hf : Integrable f μ
hf' : ¬f =ᵐ[μ] 0
this : CompleteSpace F'
⊢ AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) (Integrable.toL1 f hf))) μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
f : α → F'
s : Set α
inst✝ : CompleteSpace F'
hf : Integrable f μ
hs : MeasurableSet s
⊢ ∫ (x : α) in s, ↑↑(Integrable.toL1 f hf) x ∂μ = ∫ (x : α) in s, f x ∂μ

Tactic:
]

Goal state after:
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
f : α → F'
s : Set α
inst✝ : CompleteSpace F'
hf : Integrable f μ
hs : MeasurableSet s
⊢ ∫ (x : α) in s, ↑↑(Integrable.toL1 f hf) x ∂μ = ∫ (x : α) in s, f x ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
ι : Type u_5
inst✝ : Countable ι
f : ι → α → E
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
⊢ ↑↑(condExpL1 hm μ fun a ↦ ∑' (i : ι), f i a) =ᵐ[μ] fun a ↦ ∑' (i : ι), μ[f i | m] a

Tactic:
]

Goal state after:
case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
ι : Type u_5
inst✝ : Countable ι
f : ι → α → E
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
⊢ ↑↑(condExpL1 hm μ fun a ↦ ∑' (i : ι), f i a) =ᵐ[μ] fun a ↦ ∑' (i : ι), μ[f i | m] a
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
f : α → F'
inst✝ : CompleteSpace F'
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ ↑↑((condExpL1CLM F' hm μ) (Integrable.toL1 f hfi)) =ᵐ[μ] f

Tactic:
]

Goal state after:
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
f : α → F'
inst✝ : CompleteSpace F'
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ ↑↑((condExpL1CLM F' hm μ) (Integrable.toL1 f hfi)) =ᵐ[μ] f
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)
t : Set α
ht : MeasurableSet t
a✝ : μ t < ∞
⊢ ∫ (x : α) in t ∩ s, (μ.restrict s)[f | m] x ∂μ = ∫ (x : α) in t ∩ s, (μ.restrict s)[f | m] x ∂μ

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
f : α → F'
inst✝ : CompleteSpace F'
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ ↑↑(Integrable.toL1 f hfi) =ᵐ[μ] ↑↑(Integrable.toL1 f hfi)

pp: case hfm
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
f : α → F'
inst✝ : CompleteSpace F'
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ AEStronglyMeasurable (↑↑(Integrable.toL1 f hfi)) μ

Tactic:
]

Goal state after:
case hfm
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
f : α → F'
inst✝ : CompleteSpace F'
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ AEStronglyMeasurable (↑↑(Integrable.toL1 f hfi)) μ