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

Goal state before:
pp: α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ∞
P : ↥(Lp F p μ) → Prop
h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    AEStronglyMeasurable f μ →
      AEStronglyMeasurable g μ →
        Disjoint (Function.support f) (Function.support g) →
          P (MemLp.toLp f hf) → P (MemLp.toLp g hg) → P (MemLp.toLp f hf + MemLp.toLp g hg)
h_closed : IsClosed {f | P ↑f}
f : ↥(Lp F p μ)
⊢ AEStronglyMeasurable (↑↑f) μ → P f

Tactic:
hf

Goal state after:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ∞
P : ↥(Lp F p μ) → Prop
h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    AEStronglyMeasurable f μ →
      AEStronglyMeasurable g μ →
        Disjoint (Function.support f) (Function.support g) →
          P (MemLp.toLp f hf) → P (MemLp.toLp g hg) → P (MemLp.toLp f hf + MemLp.toLp g hg)
h_closed : IsClosed {f | P ↑f}
f : ↥(Lp F p μ)
hf : AEStronglyMeasurable (↑↑f) μ
⊢ P f
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
hμ : ¬μ = 0
hν : ¬ν = 0
this : NeZero ν
ν' : Measure α := (ν univ)⁻¹ • ν
hμν' : μ ≪ ν'
⊢ (ν univ)⁻¹ ≠ 0

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
hμ : ¬μ = 0
hν : ¬ν = 0
this : NeZero ν
ν' : Measure α := (ν univ)⁻¹ • ν
hμν' : μ ≪ ν'
h : 0 ≤ ∫ (x : α), llr μ ν' x ∂μ + ν'.real univ - μ.real univ
⊢ (ν univ)⁻¹ ≠ 0

Tactic:
simp

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

Goal state before:
pp: case refine_2
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ∞
P : ↥(Lp F p μ) → Prop
h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    AEStronglyMeasurable f μ →
      AEStronglyMeasurable g μ →
        Disjoint (Function.support f) (Function.support g) →
          P (MemLp.toLp f hf) → P (MemLp.toLp g hg) → P (MemLp.toLp f hf + MemLp.toLp g hg)
h_closed : IsClosed {f | P ↑f}
f✝ : ↥(Lp F p μ)
hf : AEStronglyMeasurable (↑↑f✝) μ
f' : ↥(lpMeas F ℝ m p μ) := ⟨f✝, hf⟩
g✝ : ↥(Lp F p (μ.trim hm)) := (lpMeasToLpTrimLie F ℝ p μ hm) f'
hfg : f' = (lpMeasToLpTrimLie F ℝ p μ hm).symm g✝
f g : α → F
⊢ ∀ (hf : MemLp f p (μ.trim hm)) (hg : MemLp g p (μ.trim hm)),
    Disjoint (Function.support f) (Function.support g) →
      P ↑((lpMeasToLpTrimLie F ℝ p μ hm).symm (MemLp.toLp f hf)) →
        P ↑((lpMeasToLpTrimLie F ℝ p μ hm).symm (MemLp.toLp g hg)) →
          P ↑((lpMeasToLpTrimLie F ℝ p μ hm).symm (MemLp.toLp f hf + MemLp.toLp g hg))

Tactic:
hf

Goal state after:
case refine_2
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ∞
P : ↥(Lp F p μ) → Prop
h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    AEStronglyMeasurable f μ →
      AEStronglyMeasurable g μ →
        Disjoint (Function.support f) (Function.support g) →
          P (MemLp.toLp f hf) → P (MemLp.toLp g hg) → P (MemLp.toLp f hf + MemLp.toLp g hg)
h_closed : IsClosed {f | P ↑f}
f✝ : ↥(Lp F p μ)
hf✝ : AEStronglyMeasurable (↑↑f✝) μ
f' : ↥(lpMeas F ℝ m p μ) := ⟨f✝, hf✝⟩
g✝ : ↥(Lp F p (μ.trim hm)) := (lpMeasToLpTrimLie F ℝ p μ hm) f'
hfg : f' = (lpMeasToLpTrimLie F ℝ p μ hm).symm g✝
f g : α → F
hf : MemLp f p (μ.trim hm)
⊢ ∀ (hg : MemLp g p (μ.trim hm)),
    Disjoint (Function.support f) (Function.support g) →
      P ↑((lpMeasToLpTrimLie F ℝ p μ hm).symm (MemLp.toLp f hf)) →
        P ↑((lpMeasToLpTrimLie F ℝ p μ hm).symm (MemLp.toLp g hg)) →
          P ↑((lpMeasToLpTrimLie F ℝ p μ hm).symm (MemLp.toLp f hf + MemLp.toLp g hg))
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
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
x : G
⊢ μ ∅ ≠ ∞

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ∞
P : ↥(Lp F p μ) → Prop
h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    StronglyMeasurable f →
      StronglyMeasurable g →
        Disjoint (Function.support f) (Function.support g) →
          P (MemLp.toLp f hf) → P (MemLp.toLp g hg) → P (MemLp.toLp f hf + MemLp.toLp g hg)
h_closed : IsClosed {f | P ↑f}
f : ↥(Lp F p μ)
⊢ AEStronglyMeasurable (↑↑f) μ → P f

Tactic:
hf

Goal state after:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ∞
P : ↥(Lp F p μ) → Prop
h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    StronglyMeasurable f →
      StronglyMeasurable g →
        Disjoint (Function.support f) (Function.support g) →
          P (MemLp.toLp f hf) → P (MemLp.toLp g hg) → P (MemLp.toLp f hf + MemLp.toLp g hg)
h_closed : IsClosed {f | P ↑f}
f : ↥(Lp F p μ)
hf : AEStronglyMeasurable (↑↑f) μ
⊢ P f
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
f : ↥(Lp ℝ 2 μ)
hf : ↑↑f =ᵐ[μ.restrict s] 0
x : α
hx : ↑↑f x = 0 x
⊢ ↑‖0 x‖₊ = 0 x

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ∞
P : ↥(Lp F p μ) → Prop
h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    StronglyMeasurable f →
      StronglyMeasurable g →
        Disjoint (Function.support f) (Function.support g) →
          P (MemLp.toLp f hf) → P (MemLp.toLp g hg) → P (MemLp.toLp f hf + MemLp.toLp g hg)
h_closed : IsClosed {f | P ↑f}
f✝ : ↥(Lp F p μ)
hf : AEStronglyMeasurable (↑↑f✝) μ
f g : α → F
⊢ ∀ (hf : MemLp f p μ) (hg : MemLp g p μ),
    AEStronglyMeasurable f μ →
      AEStronglyMeasurable g μ →
        Disjoint (Function.support f) (Function.support g) →
          P (MemLp.toLp f hf) → P (MemLp.toLp g hg) → P (MemLp.toLp f hf + MemLp.toLp g hg)

Tactic:
hf

Goal state after:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ∞
P : ↥(Lp F p μ) → Prop
h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    StronglyMeasurable f →
      StronglyMeasurable g →
        Disjoint (Function.support f) (Function.support g) →
          P (MemLp.toLp f hf) → P (MemLp.toLp g hg) → P (MemLp.toLp f hf + MemLp.toLp g hg)
h_closed : IsClosed {f | P ↑f}
f✝ : ↥(Lp F p μ)
hf✝ : AEStronglyMeasurable (↑↑f✝) μ
f g : α → F
hf : MemLp f p μ
⊢ ∀ (hg : MemLp g p μ),
    AEStronglyMeasurable f μ →
      AEStronglyMeasurable g μ →
        Disjoint (Function.support f) (Function.support g) →
          P (MemLp.toLp f hf) → P (MemLp.toLp g hg) → P (MemLp.toLp f hf + MemLp.toLp g hg)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
ht : MeasurableSet t
hμt : μ t ≠ ∞
x : α
hx : ↑↑(indicatorConstLp 2 hs hμs 1) x = s.indicator (fun x ↦ 1) x
h✝ : x ∈ s
⊢ ↑‖1‖₊ = 1

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
ht : MeasurableSet t
hμt : μ t ≠ ∞
x : α
hx : ↑↑(indicatorConstLp 2 hs hμs 1) x = s.indicator (fun x ↦ 1) x
h✝ : x ∉ s
⊢ ↑‖0‖₊ = 0

Tactic:
simp

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

Goal state before:
pp: case refine_2
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)
s : Set α
inst✝ : CompleteSpace F'
f✝ : ↥(Lp F' 1 μ)
hs : MeasurableSet s
hμs : μ s ≠ ∞
f g : α → F'
hf_Lp : MemLp f 1 μ
hg_Lp : MemLp g 1 μ
a✝ : Disjoint (Function.support f) (Function.support g)
⊢ ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf_Lp)) x ∂μ = ∫ (x : α) in s, ↑↑(MemLp.toLp f hf_Lp) x ∂μ →
    ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) (MemLp.toLp g hg_Lp)) x ∂μ = ∫ (x : α) in s, ↑↑(MemLp.toLp g hg_Lp) x ∂μ →
      ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf_Lp + MemLp.toLp g hg_Lp)) x ∂μ =
        ∫ (x : α) in s, ↑↑(MemLp.toLp f hf_Lp + MemLp.toLp g hg_Lp) x ∂μ

Tactic:
hf

Goal state after:
case refine_2
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)
s : Set α
inst✝ : CompleteSpace F'
f✝ : ↥(Lp F' 1 μ)
hs : MeasurableSet s
hμs : μ s ≠ ∞
f g : α → F'
hf_Lp : MemLp f 1 μ
hg_Lp : MemLp g 1 μ
a✝ : Disjoint (Function.support f) (Function.support g)
hf : ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf_Lp)) x ∂μ = ∫ (x : α) in s, ↑↑(MemLp.toLp f hf_Lp) x ∂μ
⊢ ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) (MemLp.toLp g hg_Lp)) x ∂μ = ∫ (x : α) in s, ↑↑(MemLp.toLp g hg_Lp) x ∂μ →
    ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf_Lp + MemLp.toLp g hg_Lp)) x ∂μ =
      ∫ (x : α) in s, ↑↑(MemLp.toLp f hf_Lp + MemLp.toLp g hg_Lp) x ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ∞
P : (α → F) → Prop
h_ind : ∀ (c : F) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → P (s.indicator fun x ↦ c)
h_add :
  ∀ ⦃f g : α → F⦄,
    Disjoint (Function.support f) (Function.support g) →
      MemLp f p μ → MemLp g p μ → StronglyMeasurable f → StronglyMeasurable g → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑↑f}
h_ae : ∀ ⦃f g : α → F⦄, f =ᵐ[μ] g → MemLp f p μ → P f → P g
f : α → F
⊢ MemLp f p μ → AEStronglyMeasurable f μ → P f

Tactic:
hf

Goal state after:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ∞
P : (α → F) → Prop
h_ind : ∀ (c : F) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → P (s.indicator fun x ↦ c)
h_add :
  ∀ ⦃f g : α → F⦄,
    Disjoint (Function.support f) (Function.support g) →
      MemLp f p μ → MemLp g p μ → StronglyMeasurable f → StronglyMeasurable g → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑↑f}
h_ae : ∀ ⦃f g : α → F⦄, f =ᵐ[μ] g → MemLp f p μ → P f → P g
f : α → F
hf : MemLp f p μ
⊢ AEStronglyMeasurable f μ → P f
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: case refine_2
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)
inst✝ : CompleteSpace F'
f✝ : ↥(Lp F' 1 μ)
f g : α → F'
⊢ ∀ (hf : MemLp f 1 μ) (hg : MemLp g 1 μ),
    Disjoint (Function.support f) (Function.support g) →
      AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf))) μ →
        AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) (MemLp.toLp g hg))) μ →
          AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf + MemLp.toLp g hg))) μ

Tactic:
hf

Goal state after:
case refine_2
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)
inst✝ : CompleteSpace F'
f✝ : ↥(Lp F' 1 μ)
f g : α → F'
hf : MemLp f 1 μ
⊢ ∀ (hg : MemLp g 1 μ),
    Disjoint (Function.support f) (Function.support g) →
      AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf))) μ →
        AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) (MemLp.toLp g hg))) μ →
          AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) (MemLp.toLp f hf + MemLp.toLp g hg))) μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: case refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
inst✝ : SigmaFinite (μ.trim hm)
h : AEStronglyMeasurable (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) μ
⊢ ∀ (s_1 : Set α),
    MeasurableSet s_1 →
      (μ.trim hm) s_1 < ∞ →
        IntegrableOn (AEStronglyMeasurable.mk (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) h) s_1 (μ.trim hm)

pp: case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
inst✝ : SigmaFinite (μ.trim hm)
h : AEStronglyMeasurable (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) μ
⊢ ∀ (s_1 : Set α),
    MeasurableSet s_1 →
      (μ.trim hm) s_1 < ∞ →
        0 ≤
          ∫ (x : α) in s_1,
            AEStronglyMeasurable.mk (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) h x ∂μ.trim hm

Tactic:
· rintro t - -
    refine @Integrable.integrableOn _ _ m _ _ _ _ _ ?_
    refine Integrable.trim hm ?_ h.stronglyMeasurable_mk
    rw [integrable_congr h.ae_eq_mk.symm]
    exact integrable_condExpL2_indicator hm hs hμs _

Goal state after:
case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
inst✝ : SigmaFinite (μ.trim hm)
h : AEStronglyMeasurable (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) μ
⊢ ∀ (s_1 : Set α),
    MeasurableSet s_1 →
      (μ.trim hm) s_1 < ∞ →
        0 ≤
          ∫ (x : α) in s_1,
            AEStronglyMeasurable.mk (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) h x ∂μ.trim hm
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
⊢ 0 =ᵐ[μ] 0 + 0

Tactic:
simp

Goal state after:
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)
⊢ 0 =ᵐ[μ] 0 + 0

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
inst✝ : SigmaFinite (μ.trim hm)
h : AEStronglyMeasurable (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) μ
t : Set α
ht : MeasurableSet t
hμt : (μ.trim hm) t < ∞
⊢ ∀ᵐ (x : α) ∂μ,
    x ∈ t →
      AEStronglyMeasurable.mk (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) h x =
        ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) x

Tactic:
filter_upwards [h.ae_eq_mk] with x hx using fun _ => hx.symm

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

Goal state before:
pp: case refine_2
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)
inst✝ : CompleteSpace F'
f✝ : ↥(lpMeas F' ℝ m 1 μ)
g✝ : ↥(Lp F' 1 (μ.trim hm)) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f✝
hfg : f✝ = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g✝
f g : α → F'
⊢ ∀ (hf : MemLp f 1 (μ.trim hm)) (hg : MemLp g 1 (μ.trim hm)),
    Disjoint (Function.support f) (Function.support g) →
      (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf)) =
          ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf)) →
        (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp g hg)) =
            ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp g hg)) →
          (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf + MemLp.toLp g hg)) =
            ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf + MemLp.toLp g hg))

Tactic:
hf

Goal state after:
case refine_2
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)
inst✝ : CompleteSpace F'
f✝ : ↥(lpMeas F' ℝ m 1 μ)
g✝ : ↥(Lp F' 1 (μ.trim hm)) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f✝
hfg : f✝ = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g✝
f g : α → F'
hf : MemLp f 1 (μ.trim hm)
⊢ ∀ (hg : MemLp g 1 (μ.trim hm)),
    Disjoint (Function.support f) (Function.support g) →
      (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf)) =
          ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf)) →
        (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp g hg)) =
            ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp g hg)) →
          (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf + MemLp.toLp g hg)) =
            ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf + MemLp.toLp g hg))
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
⊢ 0 =ᵐ[μ] c • 0

Tactic:
simp

Goal state after:
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)
⊢ 0 =ᵐ[μ] c • 0

Tactic:
simp

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:
simp

Goal state after:
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:
suffices h_int_restrict : Integrable (t.indicator ((μ.restrict s)[f | m])) (μ.restrict s) by
      rw [integrable_indicator_iff (hm _ hs_m), IntegrableOn]
      exact h_int_restrict

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 (t.indicator (μ.restrict s)[f | m]) (μ.restrict s)
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
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)
A : ∀ᵐ (a : α) ∂μ, ∀ (i : ι), μ[f i | m] a = ↑↑(condExpL1 hm μ (f i)) a
i : ι
⊢ 0 ≤ 1

Tactic:
simp

Goal state after:
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
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)
A : ∀ᵐ (a : α) ∂μ, ∀ (i : ι), μ[f i | m] a = ↑↑(condExpL1 hm μ (f i)) a
i : ι
⊢ ↑(NNReal.mk 1 ⋯) * ∫⁻ (x : α), ‖f i x‖ₑ ∂μ = ∫⁻ (a : α), ‖f i a‖ₑ ∂μ

Tactic:
simp

Goal state after: