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

Goal state before:
pp: case refine_1
α : Type u_1
E' : Type u_2
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn (↑↑↑f) s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
g : α → E'
hg_sm : FinStronglyMeasurable g (μ.trim hm)
hfg : ↑↑↑f =ᵐ[μ] g
⊢ ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ

pp: case refine_2
α : Type u_1
E' : Type u_2
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn (↑↑↑f) s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
g : α → E'
hg_sm : FinStronglyMeasurable g (μ.trim hm)
hfg : ↑↑↑f =ᵐ[μ] g
⊢ ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, g x ∂μ = 0

Tactic:
· intro s hs hμs
    have hfg_restrict : f =ᵐ[μ.restrict s] g := ae_restrict_of_ae hfg
    rw [IntegrableOn, integrable_congr hfg_restrict.symm]
    exact hf_int_finite s hs hμs

Goal state after:
case refine_2
α : Type u_1
E' : Type u_2
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn (↑↑↑f) s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
g : α → E'
hg_sm : FinStronglyMeasurable g (μ.trim hm)
hfg : ↑↑↑f =ᵐ[μ] g
⊢ ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, g x ∂μ = 0
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : μ s ≠ ∞
hμt : μ t ≠ ∞
hst : Disjoint s t
x : G
⊢ ↑↑(condExpIndSMul hm ⋯ ⋯ x) =ᵐ[μ] ↑↑(condExpIndSMul hm hs hμs x) + ↑↑(condExpIndSMul hm ht hμt x)

Tactic:
condExpIndSMul

Goal state after:
α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : μ s ≠ ∞
hμt : μ t ≠ ∞
hst : Disjoint s t
x : G
⊢ ↑↑((compLpL 2 μ (toSpanSingleton ℝ x)) ↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 ⋯ ⋯ 1))) =ᵐ[μ]
    ↑↑(condExpIndSMul hm hs hμs x) + ↑↑(condExpIndSMul hm ht hμt x)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hf : ¬Integrable f μ
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ μ[f | m] = 0

Tactic:
condExp_of_not_sigmaFinite hm hμm

Goal state after:
case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hf : ¬Integrable f μ
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ 0 = 0
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ μ[0 | m] = 0

Tactic:
condExp_of_not_sigmaFinite hm hμm

Goal state after:
case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ 0 = 0
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ StronglyMeasurable μ[f | m]

Tactic:
condExp_of_not_sigmaFinite hm hμm

Goal state after:
case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ StronglyMeasurable 0
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 (fun a ↦ ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) a).toReal * log ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) a).toReal) (ν ⊗ₘ η)
hμν_ac : μ ≪ ν
hκη_ac : μ ⊗ₘ κ ≪ μ ⊗ₘ η
⊢ Integrable (llr μ ν) μ

Tactic:
replace h_int := convexOn_mul_log.integrable_apply_rnDeriv_of_integrable_compProd
    continuous_mul_log.stronglyMeasurable continuous_mul_log.continuousWithinAt h_int hκη_ac

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
hμν_ac : μ ≪ ν
hκη_ac : μ ⊗ₘ κ ≪ μ ⊗ₘ η
h_int : Integrable (fun a ↦ ((∂μ/∂ν) a).toReal * log ((∂μ/∂ν) a).toReal) ν
⊢ Integrable (llr μ ν) μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
h : f =ᵐ[μ] g
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ μ[f | m] =ᵐ[μ] μ[g | m]

Tactic:
condExp_of_not_sigmaFinite hm hμm

Goal state after:
case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
h : f =ᵐ[μ] g
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ 0 =ᵐ[μ] 0
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : NormedSpace ℝ G
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : G
h : AEStronglyMeasurable (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) μ
⊢ AEStronglyMeasurable (↑↑(condExpIndSMul hm hs hμs x)) μ

Tactic:
condExpIndSMul

Goal state after:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : NormedSpace ℝ G
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : G
h : AEStronglyMeasurable (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) μ
⊢ AEStronglyMeasurable (↑↑((compLpL 2 μ (toSpanSingleton ℝ x)) ↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)))) μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ Integrable μ[f | m] μ

Tactic:
condExp_of_not_sigmaFinite hm hμm

Goal state after:
case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ Integrable 0 μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : NormedSpace ℝ G
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
x y : G
⊢ condExpIndSMul hm hs hμs (x + y) = condExpIndSMul hm hs hμs x + condExpIndSMul hm hs hμs y

Tactic:
condExpIndSMul

Goal state after:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : NormedSpace ℝ G
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
x y : G
⊢ (compLpL 2 μ (toSpanSingleton ℝ (x + y))) ↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) =
    (compLpL 2 μ (toSpanSingleton ℝ x)) ↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) +
      (compLpL 2 μ (toSpanSingleton ℝ y)) ↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: 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
hm : m ≤ m₀
inst✝ : SigmaFinite (μ.trim hm)
f g : α → E
hf : Integrable f μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, f x ∂μ
hgm : AEStronglyMeasurable g μ
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, μ[f | m] x ∂μ

Tactic:
setIntegral_condExp hm hf hs

Goal state after:
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
hm : m ≤ m₀
inst✝ : SigmaFinite (μ.trim hm)
f g : α → E
hf : Integrable f μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, f x ∂μ
hgm : AEStronglyMeasurable g μ
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, f x ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case neg
α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hf : Integrable f μ
hg : Integrable g μ
m : MeasurableSpace α
hm : ¬m ≤ m₀
⊢ μ[f + g | m] =ᵐ[μ] μ[f | m] + μ[g | m]

pp: case pos
α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hf : Integrable f μ
hg : Integrable g μ
m : MeasurableSpace α
hm : m ≤ m₀
⊢ μ[f + g | m] =ᵐ[μ] μ[f | m] + μ[g | m]

Tactic:
· simp_rw [condExp_of_not_le hm]; simp

Goal state after:
case pos
α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hf : Integrable f μ
hg : Integrable g μ
m : MeasurableSpace α
hm : m ≤ m₀
⊢ μ[f + g | m] =ᵐ[μ] μ[f | m] + μ[g | m]
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_ac_μν : μ ≪ ν
h_ac_κη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
h_iff_κ : Integrable (llr μ ν) μ ↔ Integrable (fun x ↦ llr μ ν x.1) (μ ⊗ₘ κ)
h_int : Integrable (fun x ↦ log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) x).toReal) (μ ⊗ₘ κ)
h_int_iff :
  Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ) ↔
    Integrable (fun x ↦ log ((∂μ/∂ν) x.1).toReal + log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) x).toReal) (μ ⊗ₘ κ)
h_int1 : Integrable (fun x ↦ llr μ ν x.1) (μ ⊗ₘ κ)
⊢ Integrable (fun x ↦ llr μ ν x.1) (μ ⊗ₘ κ) ∧ Integrable (llr (μ ⊗ₘ κ) (μ ⊗ₘ η)) (μ ⊗ₘ κ)

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_ac_μν : μ ≪ ν
h_ac_κη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
h_iff_κ : Integrable (llr μ ν) μ ↔ Integrable (fun x ↦ llr μ ν x.1) (μ ⊗ₘ κ)
h_int : Integrable (fun x ↦ log ((∂μ/∂ν) x.1).toReal + log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) x).toReal) (μ ⊗ₘ κ)
h_int_iff :
  Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ) ↔
    Integrable (fun x ↦ log ((∂μ/∂ν) x.1).toReal + log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) x).toReal) (μ ⊗ₘ κ)
h_int1 : Integrable (fun x ↦ llr μ ν x.1) (μ ⊗ₘ κ)
⊢ Integrable (fun x ↦ log ((∂μ/∂ν) x.1).toReal) (μ ⊗ₘ κ)

Tactic:
· exact ⟨h_int1, h_int⟩

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_ac_μν : μ ≪ ν
h_ac_κη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
h_iff_κ : Integrable (llr μ ν) μ ↔ Integrable (fun x ↦ llr μ ν x.1) (μ ⊗ₘ κ)
h_int : Integrable (fun x ↦ log ((∂μ/∂ν) x.1).toReal + log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) x).toReal) (μ ⊗ₘ κ)
h_int_iff :
  Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ) ↔
    Integrable (fun x ↦ log ((∂μ/∂ν) x.1).toReal + log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) x).toReal) (μ ⊗ₘ κ)
h_int1 : Integrable (fun x ↦ llr μ ν x.1) (μ ⊗ₘ κ)
⊢ Integrable (fun x ↦ log ((∂μ/∂ν) x.1).toReal) (μ ⊗ₘ κ)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case neg
α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hf : Integrable f μ
hg : Integrable g μ
m : MeasurableSpace α
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ μ[f + g | m] =ᵐ[μ] μ[f | m] + μ[g | m]

Tactic:
condExp_of_not_sigmaFinite hm hμm

Goal state after:
case neg
α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hf : Integrable f μ
hg : Integrable g μ
m : MeasurableSpace α
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ 0 =ᵐ[μ] 0 + 0
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: case neg
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
hs : MeasurableSet s
hf : f =ᵐ[μ.restrict s] 0
hm : m ≤ m0
hμm : ¬SigmaFinite (μ.trim hm)
⊢ μ[f | m] =ᵐ[μ.restrict s] 0

Tactic:
condExp_of_not_sigmaFinite hm hμm

Goal state after:
case neg
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
hs : MeasurableSet s
hf : f =ᵐ[μ.restrict s] 0
hm : m ≤ m0
hμm : ¬SigmaFinite (μ.trim hm)
⊢ 0 =ᵐ[μ.restrict s] 0
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: case pos
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
hs : MeasurableSet s
hf : f =ᵐ[μ.restrict s] 0
hm : m ≤ m0
hμm : SigmaFinite (μ.trim hm)
⊢ μ[f | m] =ᵐ[μ.restrict s] 0

Tactic:
have : SigmaFinite (μ.trim hm) := hμm

Goal state after:
case pos
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
hs : MeasurableSet s
hf : f =ᵐ[μ.restrict s] 0
hm : m ≤ m0
hμm this : SigmaFinite (μ.trim hm)
⊢ μ[f | m] =ᵐ[μ.restrict s] 0
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case neg
α : Type u_1
E : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
m₀ : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
f : α → E
m : MeasurableSpace α
hm : ¬m ≤ m₀
⊢ μ[c • f | m] =ᵐ[μ] c • μ[f | m]

pp: case pos
α : Type u_1
E : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
m₀ : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
f : α → E
m : MeasurableSpace α
hm : m ≤ m₀
⊢ μ[c • f | m] =ᵐ[μ] c • μ[f | m]

Tactic:
· simp_rw [condExp_of_not_le hm]; simp

Goal state after:
case pos
α : Type u_1
E : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
m₀ : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
f : α → E
m : MeasurableSpace α
hm : m ≤ m₀
⊢ μ[c • f | m] =ᵐ[μ] c • μ[f | m]
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case neg
α : Type u_1
E : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
m₀ : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
f : α → E
m : MeasurableSpace α
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ μ[c • f | m] =ᵐ[μ] c • μ[f | m]

Tactic:
condExp_of_not_sigmaFinite hm hμm

Goal state after:
case neg
α : Type u_1
E : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
m₀ : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
f : α → E
m : MeasurableSpace α
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ 0 =ᵐ[μ] c • 0
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: case pos
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 : SigmaFinite (μ.trim hm)
⊢ μ[s.indicator f | m] =ᵐ[μ] s.indicator μ[f | m]

Tactic:
have : SigmaFinite (μ.trim hm) := hμm

Goal state after:
case pos
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)
⊢ μ[s.indicator f | m] =ᵐ[μ] s.indicator μ[f | m]
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∞
hXY : X =ᵐ[P] Y
⊢ P⁻[X | mΩ] =ᵐ[P] P⁻[Y | mΩ]

Tactic:
by
  by_cases hm : mΩ ≤ mΩ₀
  · by_cases hσ : SigmaFinite (P.trim hm)
    · refine ae_eq_condLExp _ _ _ (measurable_condLExp _ _ _) (fun s hs ↦ ?_)
      rw [setLIntegral_condLExp _ _ _ hs]
      apply setLIntegral_congr_fun_ae (hm s hs)
      filter_upwards [hXY] with _ h _ using h
    simp [condLExp_of_not_sigmaFinite hm hσ]
  simp [condLExp_of_not_le hm]

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

Goal state before:
pp: case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
fs gs : ℕ → α → E
f g : α → E
hfs_int : ∀ (n : ℕ), Integrable (fs n) μ
hgs_int : ∀ (n : ℕ), Integrable (gs n) μ
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs μ
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs μ
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n | m] =ᵐ[μ] μ[gs n | m]
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ μ[f | m] =ᵐ[μ] μ[g | m]

Tactic:
condExp_of_not_sigmaFinite hm hμm

Goal state after:
case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
fs gs : ℕ → α → E
f g : α → E
hfs_int : ∀ (n : ℕ), Integrable (fs n) μ
hgs_int : ∀ (n : ℕ), Integrable (gs n) μ
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs μ
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs μ
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n | m] =ᵐ[μ] μ[gs n | m]
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ 0 =ᵐ[μ] 0
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: case refine_2
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝¹ : IsFiniteMeasure μ
inst✝ : HereditarilyLindelofSpace E
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
L : ℕ → E →L[ℝ] ℝ
c : ℕ → ℝ
hLc1 : ∀ (i : ℕ), s.domRestrict ⇑(L i) + const (↑s) (c i) ≤ s.domRestrict φ
hLc2 : ⨆ i, s.domRestrict ⇑(L i) + const (↑s) (c i) = s.domRestrict φ
hp : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), (fun x ↦ (L i) (μ[f | m] x) + c i) a = μ[fun y ↦ (L i) (f y) + c i | m] a
i : ℕ
⊢ ⇑(L i) ∘ f + const α (c i) ≤ᵐ[μ] φ ∘ f

Tactic:
· filter_upwards [hf] with a ha using hLc1 i ⟨f a, ha⟩

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

Goal state before:
pp: case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PartialOrder E
inst✝² : ClosedIciTopology E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
hf : Integrable f μ
hg : Integrable g μ
hfg : f ≤ᵐ[μ] g
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ μ[f | m] ≤ᵐ[μ] μ[g | m]

Tactic:
condExp_of_not_sigmaFinite hm hμm

Goal state after:
case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PartialOrder E
inst✝² : ClosedIciTopology E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
hf : Integrable f μ
hg : Integrable g μ
hfg : f ≤ᵐ[μ] g
hm : m ≤ m₀
hμm : ¬SigmaFinite (μ.trim hm)
⊢ 0 ≤ᵐ[μ] 0
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
hm : m ≤ m𝓧
hμ : SigmaFinite (μ.trim hm)
A : μ[fun x ↦ (ENNReal.ofReal (f x)).toReal | m] =ᵐ[μ] μ[f | m]
B : 0 ≤ᵐ[μ] μ[f | m]
g : 𝓧 → ℝ≥0∞ := fun x ↦ ENNReal.ofReal (f x)
I : ∫⁻ (x : 𝓧), g x ∂μ ≠ ∞
⊢ ∫⁻ (x : 𝓧), μ⁻[g | m] x ∂μ = ∫⁻ (x : 𝓧), g x ∂μ

Tactic:
exact lintegral_condLExp _ _ _

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

Goal state before:
pp: case pos
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
α : Type u_2
m mα : MeasurableSpace α
μ : Measure α
f : α → E
hm : m ≤ mα
hμm : SigmaFinite (μ.trim hm)
hf_int : ¬Integrable f μ
⊢ (fun x ↦ ‖μ[f | m] x‖) ≤ᵐ[μ] μ[fun x ↦ ‖f x‖ | m]

pp: case neg
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
α : Type u_2
m mα : MeasurableSpace α
μ : Measure α
f : α → E
hm : m ≤ mα
hμm : SigmaFinite (μ.trim hm)
hf_int : Integrable f μ
⊢ (fun x ↦ ‖μ[f | m] x‖) ≤ᵐ[μ] μ[fun x ↦ ‖f x‖ | m]

Tactic:
· simp only [condExp_of_not_integrable hf_int, Pi.zero_apply, norm_zero]
    apply condExp_nonneg
    filter_upwards with a; positivity

Goal state after:
case neg
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
α : Type u_2
m mα : MeasurableSpace α
μ : Measure α
f : α → E
hm : m ≤ mα
hμm : SigmaFinite (μ.trim hm)
hf_int : Integrable f μ
⊢ (fun x ↦ ‖μ[f | m] x‖) ≤ᵐ[μ] μ[fun x ↦ ‖f x‖ | m]
Module: Mathlib.LinearAlgebra.AffineSpace.Matrix

Goal state before:
pp: ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁵ : AddCommGroup V
inst✝⁴ : AffineSpace V P
inst✝³ : CommRing k
inst✝² : Module k V
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b b₂ : AffineBasis ι k P
x : P
⊢ b.coords x ᵥ* (b.toMatrix ⇑b₂)⁻¹ = b₂.coords x

Tactic:
have hu := b.isUnit_toMatrix b₂
  rw [Matrix.isUnit_iff_isUnit_det] at hu
  rw [← b.toMatrix_vecMul_coords b₂, Matrix.vecMul_vecMul, Matrix.mul_nonsing_inv _ hu,
    Matrix.vecMul_one]

Goal state after: