mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosed

Goal state before:
pp: case h.refine_1
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace.PseudoMetrizableSpace X
this : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
F : Set X
hF : IsClosed[inst✝¹] F
⊢ ∀ (n : ℕ) (x : X), ((fun n ↦ thickenedIndicator ⋯ F) n) x ≤ 1

pp: case h.refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace.PseudoMetrizableSpace X
this : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
F : Set X
hF : IsClosed[inst✝¹] F
⊢ ∀ (n : ℕ), ∀ x ∈ F, 1 ≤ ((fun n ↦ thickenedIndicator ⋯ F) n) x

pp: case h.refine_3
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace.PseudoMetrizableSpace X
this : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
F : Set X
hF : IsClosed[inst✝¹] F
⊢ Tendsto (fun n x ↦ ((fun n ↦ thickenedIndicator ⋯ F) n) x) atTop (𝓝 (F.indicator fun x ↦ 1))

Tactic:
·

Goal state after:
case h.refine_1
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace.PseudoMetrizableSpace X
this : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
F : Set X
hF : IsClosed[inst✝¹] F
⊢ ∀ (n : ℕ) (x : X), ((fun n ↦ thickenedIndicator ⋯ F) n) x ≤ 1
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
f : α → E
hfint : ¬Integrable f μ
⊢ |μ[f | m]| ≤ᵐ[μ] μ[|f| | m]

pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
f : α → E
hfint : Integrable f μ
⊢ |μ[f | m]| ≤ᵐ[μ] μ[|f| | m]

Tactic:
·

Goal state after:
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
f : α → E
hfint : ¬Integrable f μ
⊢ |μ[f | m]| ≤ᵐ[μ] μ[|f| | m]
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosed

Goal state before:
pp: case h.refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace.PseudoMetrizableSpace X
this : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
F : Set X
hF : IsClosed[inst✝¹] F
⊢ ∀ (n : ℕ), ∀ x ∈ F, 1 ≤ ((fun n ↦ thickenedIndicator ⋯ F) n) x

pp: case h.refine_3
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace.PseudoMetrizableSpace X
this : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
F : Set X
hF : IsClosed[inst✝¹] F
⊢ Tendsto (fun n x ↦ ((fun n ↦ thickenedIndicator ⋯ F) n) x) atTop (𝓝 (F.indicator fun x ↦ 1))

Tactic:
·

Goal state after:
case h.refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace.PseudoMetrizableSpace X
this : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
F : Set X
hF : IsClosed[inst✝¹] F
⊢ ∀ (n : ℕ), ∀ x ∈ F, 1 ≤ ((fun n ↦ thickenedIndicator ⋯ F) n) x
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosed

Goal state before:
pp: case h.refine_3
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace.PseudoMetrizableSpace X
this : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
F : Set X
hF : IsClosed[inst✝¹] F
⊢ Tendsto (fun n x ↦ ((fun n ↦ thickenedIndicator ⋯ F) n) x) atTop (𝓝 (F.indicator fun x ↦ 1))

Tactic:
·

Goal state after:
case h.refine_3
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace.PseudoMetrizableSpace X
this : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
F : Set X
hF : IsClosed[inst✝¹] F
⊢ Tendsto (fun n x ↦ ((fun n ↦ thickenedIndicator ⋯ F) n) x) atTop (𝓝 (F.indicator fun x ↦ 1))
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
f : α → E
hm : ¬m ≤ m0
⊢ ∫ (x : α), |μ[f | m] x| ∂μ ≤ ∫ (x : α), |f x| ∂μ

pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
f : α → E
hm : m ≤ m0
⊢ ∫ (x : α), |μ[f | m] x| ∂μ ≤ ∫ (x : α), |f x| ∂μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
f : α → E
hm : ¬m ≤ m0
⊢ ∫ (x : α), |μ[f | m] x| ∂μ ≤ ∫ (x : α), |f x| ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
f : α → E
hm : m ≤ m0
hsig : ¬SigmaFinite (μ.trim hm)
⊢ ∫ (x : α), |μ[f | m] x| ∂μ ≤ ∫ (x : α), |f x| ∂μ

pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
f : α → E
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ ∫ (x : α), |μ[f | m] x| ∂μ ≤ ∫ (x : α), |f x| ∂μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
f : α → E
hm : m ≤ m0
hsig : ¬SigmaFinite (μ.trim hm)
⊢ ∫ (x : α), |μ[f | m] x| ∂μ ≤ ∫ (x : α), |f x| ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
s : Set α
hs : MeasurableSet s
f : α → E
hm : ¬m ≤ m0
⊢ ∫ (x : α) in s, |μ[f | m] x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ

pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
s : Set α
hs : MeasurableSet s
f : α → E
hm : m ≤ m0
⊢ ∫ (x : α) in s, |μ[f | m] x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
s : Set α
hs : MeasurableSet s
f : α → E
hm : ¬m ≤ m0
⊢ ∫ (x : α) in s, |μ[f | m] x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
s : Set α
hs : MeasurableSet s
f : α → E
hm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∫ (x : α) in s, |μ[f | m] x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ

pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
s : Set α
hs : MeasurableSet s
f : α → E
hm : m ≤ m0
hfint : Integrable f μ
⊢ ∫ (x : α) in s, |μ[f | m] x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
s : Set α
hs : MeasurableSet s
f : α → E
hm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∫ (x : α) in s, |μ[f | m] x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
s : Set α
hs : MeasurableSet s
f : α → E
hm : m ≤ m0
hfint : Integrable f μ
hsig : ¬SigmaFinite (μ.trim hm)
⊢ ∫ (x : α) in s, |μ[f | m] x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ

pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
s : Set α
hs : MeasurableSet s
f : α → E
hm : m ≤ m0
hfint : Integrable f μ
hsig : SigmaFinite (μ.trim hm)
⊢ ∫ (x : α) in s, |μ[f | m] x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
s : Set α
hs : MeasurableSet s
f : α → E
hm : m ≤ m0
hfint : Integrable f μ
hsig : ¬SigmaFinite (μ.trim hm)
⊢ ∫ (x : α) in s, |μ[f | m] x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosed

Goal state before:
pp: case hμν
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : HasOuterApproxClosed Ω
inst✝¹ : BorelSpace Ω
μ ν : Measure Ω
inst✝ : IsFiniteMeasure μ
h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν
key : ∀ {F : Set Ω}, IsClosed[inst✝³] F → μ F = ν F
⊢ ∀ s ∈ {s | IsClosed[inst✝³] s}, μ s = ν s

pp: case h_univ
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : HasOuterApproxClosed Ω
inst✝¹ : BorelSpace Ω
μ ν : Measure Ω
inst✝ : IsFiniteMeasure μ
h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν
key : ∀ {F : Set Ω}, IsClosed[inst✝³] F → μ F = ν F
⊢ μ univ = ν univ

pp: Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : HasOuterApproxClosed Ω
inst✝¹ : BorelSpace Ω
μ ν : Measure Ω
inst✝ : IsFiniteMeasure μ
h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν
key : ∀ {F : Set Ω}, IsClosed[inst✝³] F → μ F = ν F
⊢ inst✝⁴ = MeasurableSpace.generateFrom {s | IsClosed[inst✝³] s}

Tactic:
·

Goal state after:
case hμν
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : HasOuterApproxClosed Ω
inst✝¹ : BorelSpace Ω
μ ν : Measure Ω
inst✝ : IsFiniteMeasure μ
h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν
key : ∀ {F : Set Ω}, IsClosed[inst✝³] F → μ F = ν F
⊢ ∀ s ∈ {s | IsClosed[inst✝³] s}, μ s = ν s
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hf : 0 ≤ᵐ[μ.restrict s] f
hm : ¬m ≤ m0
⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ

pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hf : 0 ≤ᵐ[μ.restrict s] f
hm : m ≤ m0
⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hf : 0 ≤ᵐ[μ.restrict s] f
hm : ¬m ≤ m0
⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hf : 0 ≤ᵐ[μ.restrict s] f
hm : ¬m ≤ m0
⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ

pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hf : 0 ≤ᵐ[μ.restrict s] f
hm : m ≤ m0
⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ

Tactic:
· simpa [condExp_of_not_le hm] using setIntegral_nonneg_of_ae_restrict hf

Goal state after:
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hf : 0 ≤ᵐ[μ.restrict s] f
hm : m ≤ m0
⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosed

Goal state before:
pp: case h_univ
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : HasOuterApproxClosed Ω
inst✝¹ : BorelSpace Ω
μ ν : Measure Ω
inst✝ : IsFiniteMeasure μ
h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν
key : ∀ {F : Set Ω}, IsClosed[inst✝³] F → μ F = ν F
⊢ μ univ = ν univ

pp: Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : HasOuterApproxClosed Ω
inst✝¹ : BorelSpace Ω
μ ν : Measure Ω
inst✝ : IsFiniteMeasure μ
h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν
key : ∀ {F : Set Ω}, IsClosed[inst✝³] F → μ F = ν F
⊢ inst✝⁴ = MeasurableSpace.generateFrom {s | IsClosed[inst✝³] s}

Tactic:
·

Goal state after:
case h_univ
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : HasOuterApproxClosed Ω
inst✝¹ : BorelSpace Ω
μ ν : Measure Ω
inst✝ : IsFiniteMeasure μ
h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν
key : ∀ {F : Set Ω}, IsClosed[inst✝³] F → μ F = ν F
⊢ μ univ = ν univ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hf : 0 ≤ᵐ[μ.restrict s] f
hm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ

pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hf : 0 ≤ᵐ[μ.restrict s] f
hm : m ≤ m0
hfint : Integrable f μ
⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hf : 0 ≤ᵐ[μ.restrict s] f
hm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hf : 0 ≤ᵐ[μ.restrict s] f
hm : m ≤ m0
hfint : Integrable f μ
hsig : ¬SigmaFinite (μ.trim hm)
⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ

pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hf : 0 ≤ᵐ[μ.restrict s] f
hm : m ≤ m0
hfint : Integrable f μ
hsig : SigmaFinite (μ.trim hm)
⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hf : 0 ≤ᵐ[μ.restrict s] f
hm : m ≤ m0
hfint : Integrable f μ
hsig : ¬SigmaFinite (μ.trim hm)
⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosed

Goal state before:
pp: Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : HasOuterApproxClosed Ω
inst✝¹ : BorelSpace Ω
μ ν : Measure Ω
inst✝ : IsFiniteMeasure μ
h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν
key : ∀ {F : Set Ω}, IsClosed[inst✝³] F → μ F = ν F
⊢ inst✝⁴ = MeasurableSpace.generateFrom {s | IsClosed[inst✝³] s}

Tactic:
·

Goal state after:
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : HasOuterApproxClosed Ω
inst✝¹ : BorelSpace Ω
μ ν : Measure Ω
inst✝ : IsFiniteMeasure μ
h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν
key : ∀ {F : Set Ω}, IsClosed[inst✝³] F → μ F = ν F
⊢ inst✝⁴ = MeasurableSpace.generateFrom {s | IsClosed[inst✝³] s}
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
R : E
f : α → E
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ R
hn : {x | |f x| ≤ R} = ∅
⊢ ∀ᵐ (x : α) ∂μ, |μ[f | m] x| ≤ R

pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
R : E
f : α → E
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ R
hn : {x | |f x| ≤ R}.Nonempty
⊢ ∀ᵐ (x : α) ∂μ, |μ[f | m] x| ≤ R

Tactic:
·

Goal state after:
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
R : E
f : α → E
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ R
hn : {x | |f x| ≤ R} = ∅
⊢ ∀ᵐ (x : α) ∂μ, |μ[f | m] x| ≤ R
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : ¬m ≤ m0
⊢ ∀ᵐ (x : α) ∂μ, |μ[f | m] x| ≤ ↑R

pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
⊢ ∀ᵐ (x : α) ∂μ, |μ[f | m] x| ≤ ↑R

Tactic:
·

Goal state after:
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : ¬m ≤ m0
⊢ ∀ᵐ (x : α) ∂μ, |μ[f | m] x| ≤ ↑R
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∀ᵐ (x : α) ∂μ, |μ[f | m] x| ≤ ↑R

pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
⊢ ∀ᵐ (x : α) ∂μ, |μ[f | m] x| ≤ ↑R

Tactic:
·

Goal state after:
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∀ᵐ (x : α) ∂μ, |μ[f | m] x| ≤ ↑R
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos.refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |μ[f | m] x|}
⊢ IntegrableOn (fun x ↦ |μ[f | m] x|) {x | ↑R < |μ[f | m] x|} μ

pp: case pos.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |μ[f | m] x|}
⊢ ∫ (x : α) in {x | ↑R < |μ[f | m] x|}, |μ[f | m] x| ∂μ ≤ μ.real {x | ↑R < |μ[f | m] x|} * ↑R

Tactic:
·

Goal state after:
case pos.refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |μ[f | m] x|}
⊢ IntegrableOn (fun x ↦ |μ[f | m] x|) {x | ↑R < |μ[f | m] x|} μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos.refine_2.refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |μ[f | m] x|}
⊢ MeasurableSet {x | ↑R < |μ[f | m] x|}

pp: case pos.refine_2.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |μ[f | m] x|}
⊢ ∫ (x : α) in {x | ↑R < |μ[f | m] x|}, |f x| ∂μ ≤ μ.real {x | ↑R < |μ[f | m] x|} * ↑R

Tactic:
·

Goal state after:
case pos.refine_2.refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |μ[f | m] x|}
⊢ MeasurableSet {x | ↑R < |μ[f | m] x|}
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
⊢ UnifIntegrable f p μ

Tactic:
set g : ι → α → β := fun i => (hf i).choose

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i ↦ Exists.choose ⋯
⊢ UnifIntegrable f p μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hfint : ¬Integrable f μ
⊢ ∫ (x : α), ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α), ‖f x‖ ∂μ

pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hfint : Integrable f μ
⊢ ∫ (x : α), ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α), ‖f x‖ ∂μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hfint : ¬Integrable f μ
⊢ ∫ (x : α), ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α), ‖f x‖ ∂μ