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

Goal state before:
pp: α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
⊢ AEStronglyMeasurable.mk f hfm =ᵐ[μ.trim hm] AEStronglyMeasurable.mk g hgm

Tactic:
]

Goal state after:
α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
⊢ AEStronglyMeasurable.mk f hfm =ᵐ[μ.trim hm] AEStronglyMeasurable.mk g hgm
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique

Goal state before:
pp: α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ IntegrableOn (AEStronglyMeasurable.mk f hfm) s (μ.trim hm)

Tactic:
]

Goal state after:
α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ IntegrableOn (AEStronglyMeasurable.mk f hfm) s (μ.trim hm)
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : ¬Integrable (llr μ ν) μ
h_contra : Integrable (llr μ (c • ν)) μ
⊢ Integrable (llr μ (c⁻¹ • c • ν)) μ

Tactic:
]

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : ¬Integrable (llr μ ν) μ
h_contra : Integrable (llr μ (c • ν)) μ
⊢ Integrable (llr μ (c⁻¹ • c • ν)) μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique

Goal state before:
pp: α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ Integrable (AEStronglyMeasurable.mk f hfm) ((μ.restrict s).trim hm)

Tactic:
]

Goal state after:
α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ Integrable (AEStronglyMeasurable.mk f hfm) ((μ.restrict s).trim hm)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique

Goal state before:
pp: α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
hf_mk_int_finite :
  ∀ (s : Set α), MeasurableSet s → (μ.trim hm) s < ∞ → IntegrableOn (AEStronglyMeasurable.mk f hfm) s (μ.trim hm)
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ IntegrableOn (AEStronglyMeasurable.mk g hgm) s (μ.trim hm)

Tactic:
]

Goal state after:
α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
hf_mk_int_finite :
  ∀ (s : Set α), MeasurableSet s → (μ.trim hm) s < ∞ → IntegrableOn (AEStronglyMeasurable.mk f hfm) s (μ.trim hm)
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ IntegrableOn (AEStronglyMeasurable.mk g hgm) s (μ.trim hm)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique

Goal state before:
pp: α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
hf_mk_int_finite :
  ∀ (s : Set α), MeasurableSet s → (μ.trim hm) s < ∞ → IntegrableOn (AEStronglyMeasurable.mk f hfm) s (μ.trim hm)
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ Integrable (AEStronglyMeasurable.mk g hgm) ((μ.restrict s).trim hm)

Tactic:
]

Goal state after:
α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
hf_mk_int_finite :
  ∀ (s : Set α), MeasurableSet s → (μ.trim hm) s < ∞ → IntegrableOn (AEStronglyMeasurable.mk f hfm) s (μ.trim hm)
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ Integrable (AEStronglyMeasurable.mk g hgm) ((μ.restrict s).trim hm)
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : ¬Integrable (llr μ ν) μ
h_contra : Integrable (llr μ (c • ν)) μ
this : IsFiniteMeasure (↑c • ν)
h : llr μ (c⁻¹ • c • ν) =ᵐ[μ] fun x ↦ llr μ (c • ν) x + log ↑c
⊢ Integrable (fun x ↦ llr μ (c • ν) x + log ↑c) μ

Tactic:
]

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : ¬Integrable (llr μ ν) μ
h_contra : Integrable (llr μ (c • ν)) μ
this : IsFiniteMeasure (↑c • ν)
h : llr μ (c⁻¹ • c • ν) =ᵐ[μ] fun x ↦ llr μ (c • ν) x + log ↑c
⊢ Integrable (fun x ↦ llr μ (c • ν) x + log ↑c) μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique

Goal state before:
pp: α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
hf_mk_int_finite :
  ∀ (s : Set α), MeasurableSet s → (μ.trim hm) s < ∞ → IntegrableOn (AEStronglyMeasurable.mk f hfm) s (μ.trim hm)
hg_mk_int_finite :
  ∀ (s : Set α), MeasurableSet s → (μ.trim hm) s < ∞ → IntegrableOn (AEStronglyMeasurable.mk g hgm) s (μ.trim hm)
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ∫ (x : α) in s, AEStronglyMeasurable.mk f hfm x ∂μ.trim hm =
    ∫ (x : α) in s, AEStronglyMeasurable.mk g hgm x ∂μ.trim hm

Tactic:
]

Goal state after:
α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
hf_mk_int_finite :
  ∀ (s : Set α), MeasurableSet s → (μ.trim hm) s < ∞ → IntegrableOn (AEStronglyMeasurable.mk f hfm) s (μ.trim hm)
hg_mk_int_finite :
  ∀ (s : Set α), MeasurableSet s → (μ.trim hm) s < ∞ → IntegrableOn (AEStronglyMeasurable.mk g hgm) s (μ.trim hm)
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ∫ (x : α) in s, AEStronglyMeasurable.mk f hfm x ∂μ.trim hm =
    ∫ (x : α) in s, AEStronglyMeasurable.mk g hgm x ∂μ.trim hm
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
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (fun x ↦ llr μ ν x + log ↑c⁻¹) μ

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (fun x ↦ llr μ ν x + log ↑c⁻¹) μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique

Goal state before:
pp: α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
hf_mk_int_finite :
  ∀ (s : Set α), MeasurableSet s → (μ.trim hm) s < ∞ → IntegrableOn (AEStronglyMeasurable.mk f hfm) s (μ.trim hm)
hg_mk_int_finite :
  ∀ (s : Set α), MeasurableSet s → (μ.trim hm) s < ∞ → IntegrableOn (AEStronglyMeasurable.mk g hgm) s (μ.trim hm)
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ∫ (a : α) in s, f a ∂μ = ∫ (a : α) in s, g a ∂μ

Tactic:
]

Goal state after:
α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
hf_mk_int_finite :
  ∀ (s : Set α), MeasurableSet s → (μ.trim hm) s < ∞ → IntegrableOn (AEStronglyMeasurable.mk f hfm) s (μ.trim hm)
hg_mk_int_finite :
  ∀ (s : Set α), MeasurableSet s → (μ.trim hm) s < ∞ → IntegrableOn (AEStronglyMeasurable.mk g hgm) s (μ.trim hm)
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ∫ (a : α) in s, f a ∂μ = ∫ (a : α) in s, g a ∂μ
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
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : Integrable (llr μ ν) μ
h_int_left : Integrable (llr (c⁻¹ • μ) ν) (c⁻¹ • μ)
⊢ Integrable (fun x ↦ llr μ ν x - log ↑c) μ

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : Integrable (llr μ ν) μ
h_int_left : Integrable (llr (c⁻¹ • μ) ν) (c⁻¹ • μ)
⊢ Integrable (fun x ↦ llr μ ν x - log ↑c) μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
f : ↥(Lp F p μ)
⊢ AEStronglyMeasurable (↑↑f) μ ↔ AEStronglyMeasurable (↑↑f) μ

Tactic:
]

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

Goal state before:
pp: α : Type u_1
F : Type u_2
𝕜 : Type u_3
p : ℝ≥0∞
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
m m0 : MeasurableSpace α
μ : Measure α
f : ↥(Lp F p μ)
⊢ AEStronglyMeasurable (↑↑f) μ ↔ AEStronglyMeasurable (↑↑f) μ

Tactic:
]

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✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(Lp F p μ)
hf_meas : f ∈ lpMeasSubgroup F m p μ
hf : AEStronglyMeasurable (↑↑f) μ
⊢ eLpNorm (↑↑f) p μ < ∞

Tactic:
]

Goal state after:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(Lp F p μ)
hf_meas : f ∈ lpMeasSubgroup F m p μ
hf : AEStronglyMeasurable (↑↑f) μ
⊢ eLpNorm (↑↑f) p μ < ∞
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : Integrable (llr μ ν) μ
h_int_left : Integrable (llr (c⁻¹ • μ) ν) (c⁻¹ • μ)
h_int_right : Integrable (llr μ (c • ν)) μ
⊢ ENNReal.ofReal (↑c * (klDiv (c⁻¹ • μ) ν).toReal) = ↑c * klDiv (c⁻¹ • μ) ν

Tactic:
]

Goal state after:
case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : Integrable (llr μ ν) μ
h_int_left : Integrable (llr (c⁻¹ • μ) ν) (c⁻¹ • μ)
h_int_right : Integrable (llr μ (c • ν)) μ
⊢ ENNReal.ofReal (↑c * (klDiv (c⁻¹ • μ) ν).toReal) = ↑c * klDiv (c⁻¹ • μ) ν
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
f g : α → ℝ
hf : StronglyMeasurable f
hfi : IntegrableOn f s μ
hg : StronglyMeasurable g
hgi : IntegrableOn g s μ
hgf : ∀ (t : Set α), MeasurableSet t → μ t < ∞ → ∫ (x : α) in t, g x ∂μ = ∫ (x : α) in t, f x ∂μ
hs : MeasurableSet s
hμs : μ s ≠ ∞
⊢ ∫ (x : α) in {x | 0 ≤ g x}, g x ∂μ.restrict s - ∫ (x : α) in {x | g x ≤ 0}, g x ∂μ.restrict s ≤
    ∫ (x : α) in {x | 0 ≤ f x}, f x ∂μ.restrict s - ∫ (x : α) in {x | f x ≤ 0}, f x ∂μ.restrict s

Tactic:
]

Goal state after:
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
f g : α → ℝ
hf : StronglyMeasurable f
hfi : IntegrableOn f s μ
hg : StronglyMeasurable g
hgi : IntegrableOn g s μ
hgf : ∀ (t : Set α), MeasurableSet t → μ t < ∞ → ∫ (x : α) in t, g x ∂μ = ∫ (x : α) in t, f x ∂μ
hs : MeasurableSet s
hμs : μ s ≠ ∞
⊢ ∫ (x : α) in {x | 0 ≤ g x}, g x ∂μ.restrict s - ∫ (x : α) in {x | g x ≤ 0}, g x ∂μ.restrict s ≤
    ∫ (x : α) in {x | 0 ≤ f x}, f x ∂μ.restrict s - ∫ (x : α) in {x | f x ≤ 0}, f x ∂μ.restrict s
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(Lp F p (μ.trim hm))
hf_mem_ℒp : MemLp (↑↑f) p μ := memLp_of_memLp_trim hm (Lp.memLp f)
⊢ AEStronglyMeasurable (↑↑(MemLp.toLp ↑↑f ⋯)) μ

Tactic:
]

Goal state after:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(Lp F p (μ.trim hm))
hf_mem_ℒp : MemLp (↑↑f) p μ := memLp_of_memLp_trim hm (Lp.memLp f)
⊢ AEStronglyMeasurable (↑↑(MemLp.toLp ↑↑f ⋯)) μ
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : Integrable (llr μ ν) μ
h_int_left : Integrable (llr (c⁻¹ • μ) ν) (c⁻¹ • μ)
h_int_right : Integrable (llr μ (c • ν)) μ
⊢ ↑c * klDiv (c⁻¹ • μ) ν = ↑c * klDiv (c⁻¹ • μ) ν

pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : Integrable (llr μ ν) μ
h_int_left : Integrable (llr (c⁻¹ • μ) ν) (c⁻¹ • μ)
h_int_right : Integrable (llr μ (c • ν)) μ
⊢ klDiv (c⁻¹ • μ) ν ≠ ∞

Tactic:
]

Goal state after:
case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : Integrable (llr μ ν) μ
h_int_left : Integrable (llr (c⁻¹ • μ) ν) (c⁻¹ • μ)
h_int_right : Integrable (llr μ (c • ν)) μ
⊢ klDiv (c⁻¹ • μ) ν ≠ ∞
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f g : ↥(lpMeasSubgroup F m p μ)
⊢ ↑↑(lpMeasSubgroupToLpTrim F p μ hm (f + g)) =ᵐ[μ.trim hm]
    ↑↑(lpMeasSubgroupToLpTrim F p μ hm f) + ↑↑(lpMeasSubgroupToLpTrim F p μ hm g)

Tactic:
]

Goal state after:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f g : ↥(lpMeasSubgroup F m p μ)
⊢ ↑↑(lpMeasSubgroupToLpTrim F p μ hm (f + g)) =ᵐ[μ.trim hm]
    ↑↑(lpMeasSubgroupToLpTrim F p μ hm f) + ↑↑(lpMeasSubgroupToLpTrim F p μ hm g)
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
c : ℝ≥0
hc : ¬c = 0
hc' : ↑c ≠ 0
μ : Measure α
⊢ μ = μ

Tactic:
]

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

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : ¬c = 0
hc' : ↑c ≠ 0
hμ_smul : ∀ (μ : Measure α), μ = c⁻¹ • c • μ
hμν : ¬μ ≪ ν
⊢ ∞ = ∞

pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : ¬c = 0
hc' : ↑c ≠ 0
hμ_smul : ∀ (μ : Measure α), μ = c⁻¹ • c • μ
hμν : ¬μ ≪ ν
⊢ ¬c • μ ≪ c • ν

Tactic:
]

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : ¬c = 0
hc' : ↑c ≠ 0
hμ_smul : ∀ (μ : Measure α), μ = c⁻¹ • c • μ
hμν : ¬μ ≪ ν
⊢ ¬c • μ ≪ c • ν
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✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f g : ↥(lpMeasSubgroup F m p μ)
⊢ ↑↑↑(f + g) =ᵐ[μ] ↑↑(↑f + ↑g)

Tactic:
]

Goal state after:
case refine_2
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f g : ↥(lpMeasSubgroup F m p μ)
⊢ ↑↑↑(f + g) =ᵐ[μ] ↑↑(↑f + ↑g)
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : ¬c = 0
hc' : ↑c ≠ 0
hμ_smul : ∀ (μ : Measure α), μ = c⁻¹ • c • μ
hμν : ¬μ ≪ ν
h_contra : c • μ ≪ c • ν
⊢ c⁻¹ • c • μ ≪ c⁻¹ • c • ν

Tactic:
]

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : ¬c = 0
hc' : ↑c ≠ 0
hμ_smul : ∀ (μ : Measure α), μ = c⁻¹ • c • μ
hμν : ¬μ ≪ ν
h_contra : c • μ ≪ c • ν
⊢ c⁻¹ • c • μ ≪ c⁻¹ • c • ν
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(lpMeasSubgroup F m p μ)
⊢ ↑↑(lpMeasSubgroupToLpTrim F p μ hm (-f)) =ᵐ[μ.trim hm] -↑↑(lpMeasSubgroupToLpTrim F p μ hm f)

Tactic:
]

Goal state after:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(lpMeasSubgroup F m p μ)
⊢ ↑↑(lpMeasSubgroupToLpTrim F p μ hm (-f)) =ᵐ[μ.trim hm] -↑↑(lpMeasSubgroupToLpTrim F p μ hm f)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(lpMeasSubgroup F m p μ)
⊢ ↑↑↑(-f) =ᵐ[μ] -↑↑↑f

Tactic:
← Lp.coeFn_neg

Goal state after:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(lpMeasSubgroup F m p μ)
⊢ ↑↑↑(-f) =ᵐ[μ] ↑↑(-↑f)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(lpMeasSubgroup F m p μ)
⊢ ↑↑↑(-f) =ᵐ[μ] ↑↑(-↑f)

Tactic:
]

Goal state after:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(lpMeasSubgroup F m p μ)
⊢ ↑↑↑(-f) =ᵐ[μ] ↑↑(-↑f)