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

Goal state before:
pp: Ω : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
F : Set Ω
F_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F
δs : ℕ → ℝ
δs_pos : ∀ (n : ℕ), 0 < δs n
δs_lim : Tendsto δs atTop (𝓝 0)
fs : ℕ → Ω → ℝ := fun n ω ↦ ↑((thickenedIndicator ⋯ F) ω)
h : Tendsto (fun n ↦ ∫⁻ (ω : Ω), ↑((thickenedIndicator ⋯ F) ω) ∂μ) atTop (𝓝 (μ F))
n : ℕ
⊢ ENNReal.ofReal (∫ (a : Ω), ↑((thickenedIndicator ⋯ F) a) ∂μ) = ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)

pp: case hfi
Ω : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
F : Set Ω
F_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F
δs : ℕ → ℝ
δs_pos : ∀ (n : ℕ), 0 < δs n
δs_lim : Tendsto δs atTop (𝓝 0)
fs : ℕ → Ω → ℝ := fun n ω ↦ ↑((thickenedIndicator ⋯ F) ω)
h : Tendsto (fun n ↦ ∫⁻ (ω : Ω), ↑((thickenedIndicator ⋯ F) ω) ∂μ) atTop (𝓝 (μ F))
n : ℕ
⊢ Integrable (fun x ↦ ↑((thickenedIndicator ⋯ F) x)) μ

Tactic:
]

Goal state after:
case hfi
Ω : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
F : Set Ω
F_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F
δs : ℕ → ℝ
δs_pos : ∀ (n : ℕ), 0 < δs n
δs_lim : Tendsto δs atTop (𝓝 0)
fs : ℕ → Ω → ℝ := fun n ω ↦ ↑((thickenedIndicator ⋯ F) ω)
h : Tendsto (fun n ↦ ∫⁻ (ω : Ω), ↑((thickenedIndicator ⋯ F) ω) ∂μ) atTop (𝓝 (μ F))
n : ℕ
⊢ Integrable (fun x ↦ ↑((thickenedIndicator ⋯ F) x)) μ
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : SignedMeasure α
f : α → ℝ
hfi : Integrable f μ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
f' : α → ℝ := AEStronglyMeasurable.mk f ⋯
hadd' : s = t + μ.withDensityᵥ f'
this : s.HaveLebesgueDecomposition μ
⊢ s = s

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosed

Goal state before:
pp: Ω : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
F : Set Ω
F_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F
δs : ℕ → ℝ
δs_pos : ∀ (n : ℕ), 0 < δs n
δs_lim : Tendsto δs atTop (𝓝 0)
fs : ℕ → Ω → ℝ := fun n ω ↦ ↑((thickenedIndicator ⋯ F) ω)
h_eq : ∀ (n : ℕ), ∫⁻ (ω : Ω), ↑((thickenedIndicator ⋯ F) ω) ∂μ = ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)
h : Tendsto (fun n ↦ ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)) atTop (𝓝 (μ F))
⊢ Tendsto (fun n ↦ ∫ (ω : Ω), ↑((thickenedIndicator ⋯ F) ω) ∂μ) atTop (𝓝 (μ F).toReal)

Tactic:
]

Goal state after:
Ω : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
F : Set Ω
F_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F
δs : ℕ → ℝ
δs_pos : ∀ (n : ℕ), 0 < δs n
δs_lim : Tendsto δs atTop (𝓝 0)
fs : ℕ → Ω → ℝ := fun n ω ↦ ↑((thickenedIndicator ⋯ F) ω)
h_eq : ∀ (n : ℕ), ∫⁻ (ω : Ω), ↑((thickenedIndicator ⋯ F) ω) ∂μ = ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)
h : Tendsto (fun n ↦ ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)) atTop (𝓝 (μ F))
⊢ Tendsto (fun n ↦ ∫ (ω : Ω), ↑((thickenedIndicator ⋯ F) ω) ∂μ) atTop (𝓝 (μ F).toReal)
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
s : SignedMeasure α
μ : Measure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ -s = -s

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
s : SignedMeasure α
μ : Measure α
inst✝ : s.HaveLebesgueDecomposition μ
r : ℝ
⊢ r • s = r • s

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosed

Goal state before:
pp: Ω : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
F : Set Ω
F_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F
δs : ℕ → ℝ
δs_pos : ∀ (n : ℕ), 0 < δs n
δs_lim : Tendsto δs atTop (𝓝 0)
fs : ℕ → Ω → ℝ := fun n ω ↦ ↑((thickenedIndicator ⋯ F) ω)
h_eq : ∀ (n : ℕ), ∫⁻ (ω : Ω), ↑((thickenedIndicator ⋯ F) ω) ∂μ = ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)
h : Tendsto (fun n ↦ ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)) atTop (𝓝 (μ F))
n : ℕ
⊢ ∫ (ω : Ω), fs n ω ∂μ = ∫ (ω : Ω), fs n ω ∂μ

pp: Ω : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
F : Set Ω
F_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F
δs : ℕ → ℝ
δs_pos : ∀ (n : ℕ), 0 < δs n
δs_lim : Tendsto δs atTop (𝓝 0)
fs : ℕ → Ω → ℝ := fun n ω ↦ ↑((thickenedIndicator ⋯ F) ω)
h_eq : ∀ (n : ℕ), ∫⁻ (ω : Ω), ↑((thickenedIndicator ⋯ F) ω) ∂μ = ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)
h : Tendsto (fun n ↦ ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)) atTop (𝓝 (μ F))
n : ℕ
⊢ 0 ≤ ∫ (ω : Ω), fs n ω ∂μ

Tactic:
]

Goal state after:
Ω : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
F : Set Ω
F_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F
δs : ℕ → ℝ
δs_pos : ∀ (n : ℕ), 0 < δs n
δs_lim : Tendsto δs atTop (𝓝 0)
fs : ℕ → Ω → ℝ := fun n ω ↦ ↑((thickenedIndicator ⋯ F) ω)
h_eq : ∀ (n : ℕ), ∫⁻ (ω : Ω), ↑((thickenedIndicator ⋯ F) ω) ∂μ = ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)
h : Tendsto (fun n ↦ ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)) atTop (𝓝 (μ F))
n : ℕ
⊢ 0 ≤ ∫ (ω : Ω), fs n ω ∂μ
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
μ : Measure α
inst✝² : s.HaveLebesgueDecomposition μ
inst✝¹ : t.HaveLebesgueDecomposition μ
inst✝ : (s + t).HaveLebesgueDecomposition μ
⊢ s + t = s + t

Tactic:
]

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

Goal state before:
pp: case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (μ.trim hm)
f : α → ℝ
hf : Integrable f μ
s : Set α
hs : MeasurableSet s
a✝ : μ s < ∞
⊢ ∫ (x : α) in s, SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) x ∂μ =
    ∫ (x : α) in s, SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) x ∂μ

pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (μ.trim hm)
f : α → ℝ
hf : Integrable f μ
s : Set α
hs : MeasurableSet s
a✝ : μ s < ∞
⊢ StronglyMeasurable (SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm))

Tactic:
]

Goal state after:
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (μ.trim hm)
f : α → ℝ
hf : Integrable f μ
s : Set α
hs : MeasurableSet s
a✝ : μ s < ∞
⊢ StronglyMeasurable (SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm))
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
μ : Measure α
inst✝¹ : s.HaveLebesgueDecomposition μ
inst✝ : t.HaveLebesgueDecomposition μ
hst : (s + -t).HaveLebesgueDecomposition μ
⊢ (s - t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ - t.rnDeriv μ

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
μ : Measure α
inst✝¹ : s.HaveLebesgueDecomposition μ
inst✝ : t.HaveLebesgueDecomposition μ
hst : (s + -t).HaveLebesgueDecomposition μ
⊢ (s - t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ - t.rnDeriv μ
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case e_f
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), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : ¬μ s = 0
⊢ s.indicator (f i) = (s ∩ {x | C ≤ ‖f i x‖₊} ∪ s ∩ {x | ‖f i x‖₊ < C}).indicator (f i)

pp: case e_f.h
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), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : ¬μ s = 0
⊢ Disjoint (s ∩ {x | C ≤ ‖f i x‖₊}) (s ∩ {x | ‖f i x‖₊ < C})

Tactic:
]

Goal state after:
case e_f
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), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : ¬μ s = 0
⊢ s.indicator (f i) = (s ∩ {x | C ≤ ‖f i x‖₊} ∪ s ∩ {x | ‖f i x‖₊ < C}).indicator (f i)

case e_f.h
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), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : ¬μ s = 0
⊢ Disjoint (s ∩ {x | C ≤ ‖f i x‖₊}) (s ∩ {x | ‖f i x‖₊ < C})
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
μ : Measure α
inst✝¹ : s.HaveLebesgueDecomposition μ
inst✝ : t.HaveLebesgueDecomposition μ
hst : (s + -t).HaveLebesgueDecomposition μ
⊢ (s + -t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ + -t.rnDeriv μ

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
μ : Measure α
inst✝¹ : s.HaveLebesgueDecomposition μ
inst✝ : t.HaveLebesgueDecomposition μ
hst : (s + -t).HaveLebesgueDecomposition μ
⊢ (s + -t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ + -t.rnDeriv μ
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
μ : Measure α
inst✝¹ : s.HaveLebesgueDecomposition μ
inst✝ : t.HaveLebesgueDecomposition μ
hst : (s + -t).HaveLebesgueDecomposition μ
⊢ s.rnDeriv μ + -t.rnDeriv μ =ᵐ[μ] s.rnDeriv μ + -t.rnDeriv μ

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
c : ComplexMeasure α
μ : Measure α
⊢ MemLp (fun x ↦ RCLike.re (c.rnDeriv μ x)) 1 μ ∧ MemLp (fun x ↦ RCLike.im (c.rnDeriv μ x)) 1 μ

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
c : ComplexMeasure α
μ : Measure α
⊢ MemLp (fun x ↦ RCLike.re (c.rnDeriv μ x)) 1 μ ∧ MemLp (fun x ↦ RCLike.im (c.rnDeriv μ x)) 1 μ
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ (c.singularPart μ) i + (μ.withDensityᵥ (c.rnDeriv μ)) i = { re := (re c) i, im := (im c) i }

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ (c.singularPart μ) i + (μ.withDensityᵥ (c.rnDeriv μ)) i = { re := (re c) i, im := (im c) i }
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ ((c.singularPart μ) i).re + (μ.withDensityᵥ fun x ↦ (c.rnDeriv μ x).re) i = { re := (re c) i, im := (im c) i }.re

pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ Integrable (fun x ↦ (c.rnDeriv μ x).re) μ

Tactic:
]

Goal state after:
case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ ((c.singularPart μ) i).re + (μ.withDensityᵥ fun x ↦ (c.rnDeriv μ x).re) i = { re := (re c) i, im := (im c) i }.re

α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ Integrable (fun x ↦ (c.rnDeriv μ x).re) μ
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ (re c) i = { re := (re c) i, im := (im c) i }.re

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosed

Goal state before:
pp: Ω : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
F : Set Ω
F_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F
δs : ℕ → ℝ
δs_pos : ∀ (n : ℕ), 0 < δs n
δs_lim : Tendsto δs atTop (𝓝 0)
fs : ℕ → Ω → ℝ := fun n ω ↦ ↑((thickenedIndicator ⋯ F) ω)
h_eq : ∀ (n : ℕ), ∫⁻ (ω : Ω), ↑((thickenedIndicator ⋯ F) ω) ∂μ = ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)
h : Tendsto (fun n ↦ ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)) atTop (𝓝 (μ F))
h_eq' : (fun n ↦ ∫ (ω : Ω), fs n ω ∂μ) = fun n ↦ (ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)).toReal
⊢ Tendsto (fun n ↦ ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)) atTop (𝓝 (μ F))

Tactic:
]

Goal state after:
Ω : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
F : Set Ω
F_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F
δs : ℕ → ℝ
δs_pos : ∀ (n : ℕ), 0 < δs n
δs_lim : Tendsto δs atTop (𝓝 0)
fs : ℕ → Ω → ℝ := fun n ω ↦ ↑((thickenedIndicator ⋯ F) ω)
h_eq : ∀ (n : ℕ), ∫⁻ (ω : Ω), ↑((thickenedIndicator ⋯ F) ω) ∂μ = ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)
h : Tendsto (fun n ↦ ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)) atTop (𝓝 (μ F))
h_eq' : (fun n ↦ ∫ (ω : Ω), fs n ω ∂μ) = fun n ↦ (ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)).toReal
⊢ Tendsto (fun n ↦ ENNReal.ofReal (∫ (ω : Ω), fs n ω ∂μ)) atTop (𝓝 (μ F))
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ ((c.singularPart μ) i).im + (μ.withDensityᵥ fun x ↦ (c.rnDeriv μ x).im) i = { re := (re c) i, im := (im c) i }.im

pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ Integrable (fun x ↦ (c.rnDeriv μ x).im) μ

Tactic:
]

Goal state after:
case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ ((c.singularPart μ) i).im + (μ.withDensityᵥ fun x ↦ (c.rnDeriv μ x).im) i = { re := (re c) i, im := (im c) i }.im

α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ Integrable (fun x ↦ (c.rnDeriv μ x).im) μ
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ (im c) i = { re := (re c) i, im := (im c) i }.im

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case e_f
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), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : ¬μ s = 0
⊢ s.indicator (f i) = s.indicator (f i)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosed

Goal state before:
pp: X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : HasOuterApproxClosed X
F : Set X
hF : IsClosed[inst✝³] F
inst✝¹ : MeasurableSpace X
inst✝ : OpensMeasurableSpace X
μ : Measure X
n : ℕ
⊢ μ F = ∫⁻ (a : X) in F, 1 ∂μ

Tactic:
]

Goal state after:
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : HasOuterApproxClosed X
F : Set X
hF : IsClosed[inst✝³] F
inst✝¹ : MeasurableSpace X
inst✝ : OpensMeasurableSpace X
μ : Measure X
n : ℕ
⊢ μ F = ∫⁻ (a : X) in F, 1 ∂μ
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case e_f.h
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), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : ¬μ s = 0
⊢ {x | C ≤ ‖f i x‖₊} ⊓ {x | ‖f i x‖₊ < C} ⊆ ⊥

Tactic:
]

Goal state after:
case e_f.h
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), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : ¬μ s = 0
⊢ {x | C ≤ ‖f i x‖₊} ⊓ {x | ‖f i x‖₊ < C} ⊆ ⊥
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: 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 : PartialOrder E
inst✝² : ClosedIciTopology E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
f : α → E
c : E
hc : 0 ≤ c
hfc : ∀ᵐ (x : α) ∂μ, f x ≤ c
hm : m ≤ m0
hfint : Integrable f μ
hsig : SigmaFinite (μ.trim hm)
n : ℕ
h1 : (μ.restrict (spanningSets (μ.trim hm) n))[f | m] =ᵐ[μ.restrict (spanningSets (μ.trim hm) n)] μ[f | m]
h2 :
  (μ.restrict (spanningSets (μ.trim hm) n))[f | m] ≤ᵐ[μ.restrict (spanningSets (μ.trim hm) n)]
    (μ.restrict (spanningSets (μ.trim hm) n))[fun x ↦ c | m]
a : α
ha : (μ.restrict (spanningSets (μ.trim hm) n))[f | m] a = μ[f | m] a
hb : (μ.restrict (spanningSets (μ.trim hm) n))[f | m] a ≤ (μ.restrict (spanningSets (μ.trim hm) n))[fun x ↦ c | m] a
⊢ (fun x ↦ c) a ≤ c

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case e_f.h
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), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : ¬μ s = 0
x : α
hx₁ : C ≤ ‖f i x‖₊
hx₂ : ‖f i x‖₊ < C
⊢ x ∈ ⊥

Tactic:
]

Goal state after:
case e_f.h
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), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : ¬μ s = 0
x : α
hx₁ : C ≤ ‖f i x‖₊
hx₂ : ‖f i x‖₊ < C
⊢ x ∈ ⊥
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case e_f.h
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), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : ¬μ s = 0
x : α
hx₁ : x ∈ {x | C ≤ ‖f i x‖₊}
hx₂ : x ∈ {x | ‖f i x‖₊ < C}
⊢ x ∈ ⊥

Tactic:
rw [Set.mem_ofPred_eq] at hx₁ hx₂

Goal state after:
case e_f.h
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), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : ¬μ s = 0
x : α
hx₁ : C ≤ ‖f i x‖₊
hx₂ : ‖f i x‖₊ < C
⊢ 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
key :
  ∀ (x : X),
    Tendsto (fun i ↦ (thickenedIndicator ⋯ F) x) atTop
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F).indicator (fun x ↦ 1) x))
⊢ ∀ (x : X), Tendsto (fun i ↦ ((fun n ↦ thickenedIndicator ⋯ F) i) x) atTop (𝓝 (F.indicator (fun x ↦ 1) x))

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
key :
  ∀ (x : X),
    Tendsto (fun i ↦ (thickenedIndicator ⋯ F) x) atTop
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F).indicator (fun x ↦ 1) x))
⊢ ∀ (x : X), Tendsto (fun i ↦ ((fun n ↦ thickenedIndicator ⋯ F) i) x) atTop (𝓝 (F.indicator (fun x ↦ 1) x))