mathlib-initiative/mathlib-tactics huggingface dataset:
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✝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)
⊢ μ[fun a ↦ ∑' (i : ι), f i a | m] =ᵐ[μ] fun a ↦ ∑' (i : ι), μ[f i | m] a

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)
⊢ μ[fun a ↦ ∑' (i : ι), f i a | m] =ᵐ[μ] fun a ↦ ∑' (i : ι), μ[f i | m] a

Tactic:
·

Goal state after:
case neg
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)
⊢ μ[fun a ↦ ∑' (i : ι), f i a | m] =ᵐ[μ] fun a ↦ ∑' (i : ι), μ[f i | m] a
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: case refine_3
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)
⊢ AEStronglyMeasurable (s.indicator (μ.restrict s)[f | m]) μ

Tactic:
·

Goal state after:
case refine_3
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)
⊢ AEStronglyMeasurable (s.indicator (μ.restrict s)[f | m]) μ
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
hm : mΩ ≤ mΩ₀
⊢ Measurable P⁻[X | mΩ]

pp: case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
hm : ¬mΩ ≤ mΩ₀
⊢ Measurable P⁻[X | mΩ]

Tactic:
·

Goal state after:
case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
hm : mΩ ≤ mΩ₀
⊢ Measurable P⁻[X | mΩ]
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

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

pp: case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
hm : mΩ ≤ mΩ₀
hσ : ¬SigmaFinite (P.trim hm)
⊢ Measurable P⁻[X | mΩ]

Tactic:
·

Goal state after:
case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ Measurable P⁻[X | mΩ]
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

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

pp: case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
hX : ¬Measurable X
⊢ Measurable P⁻[X | mΩ]

Tactic:
·

Goal state after:
case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
hX : Measurable X
⊢ Measurable P⁻[X | mΩ]
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
hm : ¬mΩ ≤ mΩ₀
⊢ Measurable P⁻[X | mΩ]

Tactic:
·

Goal state after:
case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
hm : ¬mΩ ≤ mΩ₀
⊢ Measurable P⁻[X | mΩ]
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
hm : mΩ ≤ mΩ₀
P : Measure Ω
hσ : SigmaFinite (P.trim hm)
X : Ω → ℝ≥0∞
s : Set Ω
hs : MeasurableSet s
hX : Measurable X
⊢ ∫⁻ (ω : Ω) in s, P⁻[X | mΩ] ω ∂P = ∫⁻ (ω : Ω) in s, X ω ∂P

pp: case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
hm : mΩ ≤ mΩ₀
P : Measure Ω
hσ : SigmaFinite (P.trim hm)
X : Ω → ℝ≥0∞
s : Set Ω
hs : MeasurableSet s
hX : ¬Measurable X
⊢ ∫⁻ (ω : Ω) in s, P⁻[X | mΩ] ω ∂P = ∫⁻ (ω : Ω) in s, X ω ∂P

Tactic:
·

Goal state after:
case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
hm : mΩ ≤ mΩ₀
P : Measure Ω
hσ : SigmaFinite (P.trim hm)
X : Ω → ℝ≥0∞
s : Set Ω
hs : MeasurableSet s
hX : Measurable X
⊢ ∫⁻ (ω : Ω) in s, P⁻[X | mΩ] ω ∂P = ∫⁻ (ω : Ω) in s, X ω ∂P
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: case neg
α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
f : α → E
s : Set α
m m₂ m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hm₂ : m₂ ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
inst✝ : SigmaFinite (μ.trim hm₂)
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) ↔ MeasurableSet (s ∩ t)
hs_m₂ : MeasurableSet s
hf_int : ¬Integrable f μ
⊢ s.indicator μ[f | m] =ᵐ[μ] s.indicator μ[f | m₂]

pp: case pos
α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
f : α → E
s : Set α
m m₂ m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hm₂ : m₂ ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
inst✝ : SigmaFinite (μ.trim hm₂)
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) ↔ MeasurableSet (s ∩ t)
hs_m₂ : MeasurableSet s
hf_int : Integrable f μ
⊢ s.indicator μ[f | m] =ᵐ[μ] s.indicator μ[f | m₂]

Tactic:
·

Goal state after:
case neg
α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
f : α → E
s : Set α
m m₂ m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hm₂ : m₂ ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
inst✝ : SigmaFinite (μ.trim hm₂)
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) ↔ MeasurableSet (s ∩ t)
hs_m₂ : MeasurableSet s
hf_int : ¬Integrable f μ
⊢ s.indicator μ[f | m] =ᵐ[μ] s.indicator μ[f | m₂]
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
f : Ω → ℝ≥0∞
hf : ∫⁻ (x : Ω), f x ∂P ≠ ∞
hm : ¬mΩ ≤ mΩ₀
⊢ ∀ᵐ (x : Ω) ∂P, P⁻[f | mΩ] x < ∞

pp: case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
f : Ω → ℝ≥0∞
hf : ∫⁻ (x : Ω), f x ∂P ≠ ∞
hm : mΩ ≤ mΩ₀
⊢ ∀ᵐ (x : Ω) ∂P, P⁻[f | mΩ] x < ∞

Tactic:
·

Goal state after:
case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
f : Ω → ℝ≥0∞
hf : ∫⁻ (x : Ω), f x ∂P ≠ ∞
hm : ¬mΩ ≤ mΩ₀
⊢ ∀ᵐ (x : Ω) ∂P, P⁻[f | mΩ] x < ∞
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
f : Ω → ℝ≥0∞
hf : ∫⁻ (x : Ω), f x ∂P ≠ ∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ ∀ᵐ (x : Ω) ∂P, P⁻[f | mΩ] x < ∞

pp: case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
f : Ω → ℝ≥0∞
hf : ∫⁻ (x : Ω), f x ∂P ≠ ∞
hm : mΩ ≤ mΩ₀
hσ : ¬SigmaFinite (P.trim hm)
⊢ ∀ᵐ (x : Ω) ∂P, P⁻[f | mΩ] x < ∞

Tactic:
·

Goal state after:
case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
f : Ω → ℝ≥0∞
hf : ∫⁻ (x : Ω), f x ∂P ≠ ∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ ∀ᵐ (x : Ω) ∂P, P⁻[f | mΩ] 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
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
⊢ μ[f | m] =ᵐ[μ] μ[g | m]

pp: case pos
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
⊢ μ[f | m] =ᵐ[μ] μ[g | m]

Tactic:
·

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
⊢ μ[f | m] =ᵐ[μ] μ[g | m]
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: case pos.refine_2
α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
f : α → E
s : Set α
m m₂ m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hm₂ : m₂ ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
inst✝ : SigmaFinite (μ.trim hm₂)
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) ↔ MeasurableSet (s ∩ t)
hs_m₂ : MeasurableSet s
hf_int : Integrable f μ
⊢ AEStronglyMeasurable μ[s.indicator f | m] μ

pp: case pos.refine_1
α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
f : α → E
s : Set α
m m₂ m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hm₂ : m₂ ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
inst✝ : SigmaFinite (μ.trim hm₂)
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) ↔ MeasurableSet (s ∩ t)
hs_m₂ : MeasurableSet s
hf_int : Integrable f μ
⊢ ∀ (s_1 : Set α),
    MeasurableSet s_1 →
      μ s_1 < ∞ → ∫ (x : α) in s_1, μ[s.indicator f | m] x ∂μ = ∫ (x : α) in s_1, μ[s.indicator f | m₂] x ∂μ

Tactic:
·

Goal state after:
case pos.refine_2
α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
f : α → E
s : Set α
m m₂ m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hm₂ : m₂ ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
inst✝ : SigmaFinite (μ.trim hm₂)
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) ↔ MeasurableSet (s ∩ t)
hs_m₂ : MeasurableSet s
hf_int : Integrable f μ
⊢ AEStronglyMeasurable μ[s.indicator f | m] μ
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
f : Ω → ℝ≥0∞
hf : ∫⁻ (x : Ω), f x ∂P ≠ ∞
hm : mΩ ≤ mΩ₀
hσ : ¬SigmaFinite (P.trim hm)
⊢ ∀ᵐ (x : Ω) ∂P, P⁻[f | mΩ] x < ∞

Tactic:
·

Goal state after:
case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
f : Ω → ℝ≥0∞
hf : ∫⁻ (x : Ω), f x ∂P ≠ ∞
hm : mΩ ≤ mΩ₀
hσ : ¬SigmaFinite (P.trim hm)
⊢ ∀ᵐ (x : Ω) ∂P, P⁻[f | mΩ] x < ∞
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

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

pp: case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hXY : X =ᵐ[P] Y
hm : ¬mΩ ≤ mΩ₀
⊢ P⁻[X | mΩ] =ᵐ[P] P⁻[Y | mΩ]

Tactic:
·

Goal state after:
case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hXY : X =ᵐ[P] Y
hm : mΩ ≤ mΩ₀
⊢ P⁻[X | mΩ] =ᵐ[P] P⁻[Y | mΩ]
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hXY : X =ᵐ[P] Y
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ P⁻[X | mΩ] =ᵐ[P] P⁻[Y | mΩ]

pp: case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hXY : X =ᵐ[P] Y
hm : mΩ ≤ mΩ₀
hσ : ¬SigmaFinite (P.trim hm)
⊢ P⁻[X | mΩ] =ᵐ[P] P⁻[Y | mΩ]

Tactic:
·

Goal state after:
case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hXY : X =ᵐ[P] Y
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ P⁻[X | mΩ] =ᵐ[P] P⁻[Y | mΩ]
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]

pp: case pos
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:
·

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)
⊢ μ[f | m] =ᵐ[μ] μ[g | m]
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
hm : ¬mΩ ≤ mΩ₀
⊢ P⁻[0 | mΩ] = 0

pp: case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
hm : mΩ ≤ mΩ₀
⊢ P⁻[0 | mΩ] = 0

Tactic:
·

Goal state after:
case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
hm : ¬mΩ ≤ mΩ₀
⊢ P⁻[0 | mΩ] = 0
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: case hm
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
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hf_int : Integrable f μ
hs : IsClosed s
hc : Convex ℝ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
n : ℕ
⊢ MeasurableSet (spanningSets (μ.trim hm) n)

pp: case ht
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
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hf_int : Integrable f μ
hs : IsClosed s
hc : Convex ℝ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
n : ℕ
⊢ (μ.restrict (spanningSets (μ.trim hm) n)) {x | (fun a ↦ μ[f | m] a ∈ s) x}ᶜ = 0

Tactic:
·

Goal state after:
case hm
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
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hf_int : Integrable f μ
hs : IsClosed s
hc : Convex ℝ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
n : ℕ
⊢ MeasurableSet (spanningSets (μ.trim hm) n)
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
hm : mΩ ≤ mΩ₀
hσ : ¬SigmaFinite (P.trim hm)
⊢ P⁻[0 | mΩ] = 0

pp: case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ P⁻[0 | mΩ] = 0

Tactic:
·

Goal state after:
case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
hm : mΩ ≤ mΩ₀
hσ : ¬SigmaFinite (P.trim hm)
⊢ P⁻[0 | mΩ] = 0
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case neg
Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
inst✝ : NeZero P
X : Ω → ℝ≥0∞
hP : ¬IsFiniteMeasure P
⊢ P⁻[X | ⊥] = fun x ↦ (P Set.univ)⁻¹ • ∫⁻ (ω : Ω), X ω ∂P

pp: case pos
Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
inst✝ : NeZero P
X : Ω → ℝ≥0∞
hP : IsFiniteMeasure P
⊢ P⁻[X | ⊥] = fun x ↦ (P Set.univ)⁻¹ • ∫⁻ (ω : Ω), X ω ∂P

Tactic:
·

Goal state after:
case neg
Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
inst✝ : NeZero P
X : Ω → ℝ≥0∞
hP : ¬IsFiniteMeasure P
⊢ P⁻[X | ⊥] = fun x ↦ (P Set.univ)⁻¹ • ∫⁻ (ω : Ω), X ω ∂P
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner

Goal state before:
pp: case neg
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ≥0∞
hf_meas : AEMeasurable f μ
hf : ∫⁻ (x : 𝓧), f x ∂μ ≠ ∞
hm : ¬m ≤ m𝓧
⊢ (fun x ↦ (μ⁻[f | m] x).toReal) =ᵐ[μ] μ[fun x ↦ (f x).toReal | m]

pp: case pos
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ≥0∞
hf_meas : AEMeasurable f μ
hf : ∫⁻ (x : 𝓧), f x ∂μ ≠ ∞
hm : m ≤ m𝓧
⊢ (fun x ↦ (μ⁻[f | m] x).toReal) =ᵐ[μ] μ[fun x ↦ (f x).toReal | m]

Tactic:
·

Goal state after:
case neg
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ≥0∞
hf_meas : AEMeasurable f μ
hf : ∫⁻ (x : 𝓧), f x ∂μ ≠ ∞
hm : ¬m ≤ m𝓧
⊢ (fun x ↦ (μ⁻[f | m] x).toReal) =ᵐ[μ] μ[fun x ↦ (f x).toReal | m]
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner

Goal state before:
pp: case neg
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ≥0∞
hf_meas : AEMeasurable f μ
hf : ∫⁻ (x : 𝓧), f x ∂μ ≠ ∞
hm : m ≤ m𝓧
hμ : ¬SigmaFinite (μ.trim hm)
⊢ (fun x ↦ (μ⁻[f | m] x).toReal) =ᵐ[μ] μ[fun x ↦ (f x).toReal | m]

pp: case pos
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ≥0∞
hf_meas : AEMeasurable f μ
hf : ∫⁻ (x : 𝓧), f x ∂μ ≠ ∞
hm : m ≤ m𝓧
hμ : SigmaFinite (μ.trim hm)
⊢ (fun x ↦ (μ⁻[f | m] x).toReal) =ᵐ[μ] μ[fun x ↦ (f x).toReal | m]

Tactic:
·

Goal state after:
case neg
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ≥0∞
hf_meas : AEMeasurable f μ
hf : ∫⁻ (x : 𝓧), f x ∂μ ≠ ∞
hm : m ≤ m𝓧
hμ : ¬SigmaFinite (μ.trim hm)
⊢ (fun x ↦ (μ⁻[f | m] x).toReal) =ᵐ[μ] μ[fun x ↦ (f x).toReal | m]
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner

Goal state before:
pp: case pos.refine_1
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ≥0∞
hf_meas : AEMeasurable f μ
hf : ∫⁻ (x : 𝓧), f x ∂μ ≠ ∞
hm : m ≤ m𝓧
hμ : SigmaFinite (μ.trim hm)
⊢ Integrable (fun x ↦ (f x).toReal) μ

pp: case pos.refine_2
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ≥0∞
hf_meas : AEMeasurable f μ
hf : ∫⁻ (x : 𝓧), f x ∂μ ≠ ∞
hm : m ≤ m𝓧
hμ : SigmaFinite (μ.trim hm)
⊢ ∀ (s : Set 𝓧), MeasurableSet s → μ s < ∞ → IntegrableOn (fun x ↦ (μ⁻[f | m] x).toReal) s μ

pp: case pos.refine_3
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ≥0∞
hf_meas : AEMeasurable f μ
hf : ∫⁻ (x : 𝓧), f x ∂μ ≠ ∞
hm : m ≤ m𝓧
hμ : SigmaFinite (μ.trim hm)
⊢ ∀ (s : Set 𝓧), MeasurableSet s → μ s < ∞ → ∫ (x : 𝓧) in s, (μ⁻[f | m] x).toReal ∂μ = ∫ (x : 𝓧) in s, (f x).toReal ∂μ

pp: case pos.refine_4
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ≥0∞
hf_meas : AEMeasurable f μ
hf : ∫⁻ (x : 𝓧), f x ∂μ ≠ ∞
hm : m ≤ m𝓧
hμ : SigmaFinite (μ.trim hm)
⊢ AEStronglyMeasurable (fun x ↦ (μ⁻[f | m] x).toReal) μ

Tactic:
·

Goal state after:
case pos.refine_1
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ≥0∞
hf_meas : AEMeasurable f μ
hf : ∫⁻ (x : 𝓧), f x ∂μ ≠ ∞
hm : m ≤ m𝓧
hμ : SigmaFinite (μ.trim hm)
⊢ Integrable (fun x ↦ (f x).toReal) μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: case refine_1
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
⊢ Integrable (⇑(L i) ∘ f + const α (c i)) μ

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:
·

Goal state after:
case refine_1
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
⊢ Integrable (⇑(L i) ∘ f + const α (c i)) μ
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case inl
Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
X : Ω → ℝ≥0∞
⊢ 0⁻[X | ⊥] =ᵐ[0] fun x ↦ (0 Set.univ)⁻¹ • ∫⁻ (ω : Ω), X ω ∂0

pp: case inr
Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
hP : NeZero P
⊢ P⁻[X | ⊥] =ᵐ[P] fun x ↦ (P Set.univ)⁻¹ • ∫⁻ (ω : Ω), X ω ∂P

Tactic:
·

Goal state after:
case inl
Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
X : Ω → ℝ≥0∞
⊢ 0⁻[X | ⊥] =ᵐ[0] fun x ↦ (0 Set.univ)⁻¹ • ∫⁻ (ω : Ω), X ω ∂0