mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s✝ : Set α
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_sigma_subadd : m.IsSigmaSubadditive
this : MeasurableSpace α := (inducedOuterMeasure (fun x x_1 ↦ m x) ⋯ ⋯).caratheodory
s : Set α
hs : s ∈ C
hs_meas : MeasurableSet s
⊢ m s ≤ m s

Tactic:
]

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

Goal state before:

Tactic:
μ ((μ.sigmaFiniteSetᶜ ∩ s) ∆ (μ.sigmaFiniteSetᶜ ∩ (t ∩ μ.sigmaFiniteSet)))
              ≤ μ ((μ.sigmaFiniteSetᶜ ∩ s) ∪ (μ.sigmaFiniteSetᶜ ∩ (t ∩ μ.sigmaFiniteSet))) :=
                measure_mono symmDiff_subset_union
            _ ≤ μ (μ.sigmaFiniteSetᶜ ∩ s) + μ (μ.sigmaFiniteSetᶜ ∩ (t ∩ μ.sigmaFiniteSet)) :=
                measure_union_le _ _
            _ = 0 := by
                rw [inter_comm, ← μ.restrict_apply ms, hs, ← inter_assoc, inter_comm,
                  ← inter_assoc, inter_compl_self, empty_inter, measure_empty, zero_add]

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
mα : MeasurableSpace α
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
hC_gen : mα = MeasurableSpace.generateFrom C
m_sigma_subadd : m.IsSigmaSubadditive
hs : s ∈ C
⊢ (m.measureCaratheodory hC m_sigma_subadd) s = m s

pp: case hs
α : Type u_1
C : Set (Set α)
s : Set α
mα : MeasurableSpace α
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
hC_gen : mα = MeasurableSpace.generateFrom C
m_sigma_subadd : m.IsSigmaSubadditive
hs : s ∈ C
⊢ MeasurableSet s

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
s : Set α
mα : MeasurableSpace α
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
hC_gen : mα = MeasurableSpace.generateFrom C
m_sigma_subadd : m.IsSigmaSubadditive
hs : s ∈ C
⊢ (m.measureCaratheodory hC m_sigma_subadd) s = m s

case hs
α : Type u_1
C : Set (Set α)
s : Set α
mα : MeasurableSpace α
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
hC_gen : mα = MeasurableSpace.generateFrom C
m_sigma_subadd : m.IsSigmaSubadditive
hs : s ∈ C
⊢ MeasurableSet s
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: case hs
α : Type u_1
C : Set (Set α)
s : Set α
mα : MeasurableSpace α
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
hC_gen : mα = MeasurableSpace.generateFrom C
m_sigma_subadd : m.IsSigmaSubadditive
hs : s ∈ C
⊢ MeasurableSet s

Tactic:
]

Goal state after:
case hs
α : Type u_1
C : Set (Set α)
s : Set α
mα : MeasurableSpace α
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
hC_gen : mα = MeasurableSpace.generateFrom C
m_sigma_subadd : m.IsSigmaSubadditive
hs : s ∈ C
⊢ MeasurableSet s
Module: Mathlib.MeasureTheory.SpecificCodomains.Pi

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
p : ℝ≥0∞
ι : Type u_2
inst✝¹ : Fintype ι
E : ι → Type u_3
inst✝ : (i : ι) → NormedAddCommGroup (E i)
f : X → (i : ι) → E i
hf : ∀ (i : ι), MemLp (fun x ↦ f x i) p μ
this : f = ∑ i, Pi.single i ∘ fun x ↦ f x i
⊢ MemLp (∑ i, Pi.single i ∘ fun x ↦ f x i) p μ

Tactic:
]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
p : ℝ≥0∞
ι : Type u_2
inst✝¹ : Fintype ι
E : ι → Type u_3
inst✝ : (i : ι) → NormedAddCommGroup (E i)
f : X → (i : ι) → E i
hf : ∀ (i : ι), MemLp (fun x ↦ f x i) p μ
this : f = ∑ i, Pi.single i ∘ fun x ↦ f x i
⊢ MemLp (∑ i, Pi.single i ∘ fun x ↦ f x i) p μ
Module: Mathlib.MeasureTheory.SpecificCodomains.Pi

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
ι : Type u_2
inst✝³ : Fintype ι
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
f : X → (i : ι) → E i
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
inst✝ : ∀ (i : ι), CompleteSpace (E i)
hf : ∀ (i : ι), Integrable (fun x ↦ f x i) μ
i : ι
⊢ (∫ (x : X), f x ∂μ) i = ∫ (x : X), f x i ∂μ

Tactic:
simp [← ContinuousLinearMap.proj_apply (R := ℝ) i (∫ x, f x ∂μ),
    ← ContinuousLinearMap.integral_comp_comm _ (Integrable.of_eval hf)]

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

Goal state before:
pp: case refine_2.inl
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : CountablyGenerated X
inst✝ : SFinite μ
this✝¹ : IsSeparable (μ.restrict μ.sigmaFiniteSet)
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : (μ.restrict μ.sigmaFiniteSet).MeasureDense 𝒜
ℬ : Set (Set X) := {x | ∃ s ∈ 𝒜, s ∩ μ.sigmaFiniteSet = x}
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
hs : (μ.restrict μ.sigmaFiniteSetᶜ) s = 0
this✝ : (μ.restrict μ.sigmaFiniteSet) s ≠ ∞
t : Set X
t_mem : t ∈ 𝒜
ht : (μ.restrict μ.sigmaFiniteSet) (s ∆ t) < ENNReal.ofReal ε
this : μ (s ∆ (t ∩ μ.sigmaFiniteSet) \ μ.sigmaFiniteSet) = 0
⊢ (μ.restrict μ.sigmaFiniteSet) (s ∆ t) < ENNReal.ofReal ε

Tactic:
]

Goal state after:
case refine_2.inl
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : CountablyGenerated X
inst✝ : SFinite μ
this✝¹ : IsSeparable (μ.restrict μ.sigmaFiniteSet)
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : (μ.restrict μ.sigmaFiniteSet).MeasureDense 𝒜
ℬ : Set (Set X) := {x | ∃ s ∈ 𝒜, s ∩ μ.sigmaFiniteSet = x}
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
hs : (μ.restrict μ.sigmaFiniteSetᶜ) s = 0
this✝ : (μ.restrict μ.sigmaFiniteSet) s ≠ ∞
t : Set X
t_mem : t ∈ 𝒜
ht : (μ.restrict μ.sigmaFiniteSet) (s ∆ t) < ENNReal.ofReal ε
this : μ (s ∆ (t ∩ μ.sigmaFiniteSet) \ μ.sigmaFiniteSet) = 0
⊢ (μ.restrict μ.sigmaFiniteSet) (s ∆ t) < ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.SpecificCodomains.Pi

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
p : ℝ≥0∞
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
f : X → E × F
h : MemLp (fun x ↦ (f x).1) p μ ∧ MemLp (fun x ↦ (f x).2) p μ
this : f = (⇑(AddMonoidHom.inl E F) ∘ fun x ↦ (f x).1) + ⇑(AddMonoidHom.inr E F) ∘ fun x ↦ (f x).2
⊢ MemLp ((⇑(AddMonoidHom.inl E F) ∘ fun x ↦ (f x).1) + ⇑(AddMonoidHom.inr E F) ∘ fun x ↦ (f x).2) p μ

Tactic:
]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
p : ℝ≥0∞
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
f : X → E × F
h : MemLp (fun x ↦ (f x).1) p μ ∧ MemLp (fun x ↦ (f x).2) p μ
this : f = (⇑(AddMonoidHom.inl E F) ∘ fun x ↦ (f x).1) + ⇑(AddMonoidHom.inr E F) ∘ fun x ↦ (f x).2
⊢ MemLp ((⇑(AddMonoidHom.inl E F) ∘ fun x ↦ (f x).1) + ⇑(AddMonoidHom.inr E F) ∘ fun x ↦ (f x).2) p μ
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case refine_2.inr
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : CountablyGenerated X
inst✝ : SFinite μ
this : IsSeparable (μ.restrict μ.sigmaFiniteSet)
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : (μ.restrict μ.sigmaFiniteSet).MeasureDense 𝒜
ℬ : Set (Set X) := {x | ∃ s ∈ 𝒜, s ∩ μ.sigmaFiniteSet = x}
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
hs : (μ.restrict μ.sigmaFiniteSetᶜ) s = ∞
⊢ (μ.restrict μ.sigmaFiniteSetᶜ) s ≤ μ s

Tactic:
]

Goal state after:
case refine_2.inr
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : CountablyGenerated X
inst✝ : SFinite μ
this : IsSeparable (μ.restrict μ.sigmaFiniteSet)
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : (μ.restrict μ.sigmaFiniteSet).MeasureDense 𝒜
ℬ : Set (Set X) := {x | ∃ s ∈ 𝒜, s ∩ μ.sigmaFiniteSet = x}
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
hs : (μ.restrict μ.sigmaFiniteSetᶜ) s = ∞
⊢ (μ.restrict μ.sigmaFiniteSetᶜ) s ≤ μ s
Module: Mathlib.MeasureTheory.SpecificCodomains.WithLp

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁴ : Fact (1 ≤ q)
ι : Type u_2
inst✝³ : Fintype ι
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
f : X → PiLp q E
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
inst✝ : ∀ (i : ι), CompleteSpace (E i)
hf : ∀ (i : ι), Integrable (fun x ↦ (f x).ofLp i) μ
i : ι
⊢ ∫ (x : X), (PiLp.proj q E i) (f x) ∂μ = ∫ (x : X), (f x).ofLp i ∂μ

pp: case φ_int
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁴ : Fact (1 ≤ q)
ι : Type u_2
inst✝³ : Fintype ι
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
f : X → PiLp q E
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
inst✝ : ∀ (i : ι), CompleteSpace (E i)
hf : ∀ (i : ι), Integrable (fun x ↦ (f x).ofLp i) μ
i : ι
⊢ Integrable f μ

Tactic:
]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁴ : Fact (1 ≤ q)
ι : Type u_2
inst✝³ : Fintype ι
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
f : X → PiLp q E
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
inst✝ : ∀ (i : ι), CompleteSpace (E i)
hf : ∀ (i : ι), Integrable (fun x ↦ (f x).ofLp i) μ
i : ι
⊢ ∫ (x : X), (PiLp.proj q E i) (f x) ∂μ = ∫ (x : X), (f x).ofLp i ∂μ

case φ_int
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁴ : Fact (1 ≤ q)
ι : Type u_2
inst✝³ : Fintype ι
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
f : X → PiLp q E
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
inst✝ : ∀ (i : ι), CompleteSpace (E i)
hf : ∀ (i : ι), Integrable (fun x ↦ (f x).ofLp i) μ
i : ι
⊢ Integrable f μ
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : IsSeparable μ
inst✝ : SeparableSpace E
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
⊢ SeparableSpace ↥(Lp E p μ)

Tactic:
have h𝒜₀ := Measure.MeasureDense.fin_meas h𝒜

Goal state after:
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : IsSeparable μ
inst✝ : SeparableSpace E
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
h𝒜₀ : μ.MeasureDense {s | s ∈ 𝒜 ∧ μ s ≠ ∞}
⊢ SeparableSpace ↥(Lp E p μ)
Module: Mathlib.MeasureTheory.SpecificCodomains.WithLp

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hf : Integrable f μ
⊢ (∫ (x : X), f x ∂μ).ofLp.1 = ∫ (x : X), (f x).fst ∂μ

Tactic:
]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hf : Integrable f μ
⊢ (∫ (x : X), f x ∂μ).ofLp.1 = ∫ (x : X), (f x).fst ∂μ
Module: Mathlib.MeasureTheory.SpecificCodomains.WithLp

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hf : Integrable f μ
⊢ ∫ (x : X), ((WithLp.prodContinuousLinearEquiv q ℝ E F) (f x)).1 ∂μ = ∫ (x : X), (f x).fst ∂μ

pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hf : Integrable f μ
⊢ Integrable (fun x ↦ (WithLp.prodContinuousLinearEquiv q ℝ E F) (f x)) μ

Tactic:
]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hf : Integrable f μ
⊢ ∫ (x : X), ((WithLp.prodContinuousLinearEquiv q ℝ E F) (f x)).1 ∂μ = ∫ (x : X), (f x).fst ∂μ

X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hf : Integrable f μ
⊢ Integrable (fun x ↦ (WithLp.prodContinuousLinearEquiv q ℝ E F) (f x)) μ
Module: Mathlib.MeasureTheory.SpecificCodomains.WithLp

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
hf : Integrable f μ
⊢ (∫ (x : X), f x ∂μ).ofLp.2 = ∫ (x : X), (f x).snd ∂μ

Tactic:
]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
hf : Integrable f μ
⊢ (∫ (x : X), f x ∂μ).ofLp.2 = ∫ (x : X), (f x).snd ∂μ
Module: Mathlib.MeasureTheory.SpecificCodomains.WithLp

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
hf : Integrable f μ
⊢ ∫ (x : X), ((WithLp.prodContinuousLinearEquiv q ℝ E F) (f x)).2 ∂μ = ∫ (x : X), (f x).snd ∂μ

pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
hf : Integrable f μ
⊢ Integrable (fun x ↦ (WithLp.prodContinuousLinearEquiv q ℝ E F) (f x)) μ

Tactic:
]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
hf : Integrable f μ
⊢ ∫ (x : X), ((WithLp.prodContinuousLinearEquiv q ℝ E F) (f x)).2 ∂μ = ∫ (x : X), (f x).snd ∂μ

X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
hf : Integrable f μ
⊢ Integrable (fun x ↦ (WithLp.prodContinuousLinearEquiv q ℝ E F) (f x)) μ
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case refine_2.refine_1
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : IsSeparable μ
inst✝ : SeparableSpace E
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
𝒜₀ : Set (Set X) := {s | s ∈ 𝒜 ∧ μ s ≠ ∞}
h𝒜₀ : μ.MeasureDense 𝒜₀
count_𝒜₀ : 𝒜₀.Countable
p_ne_zero : p ≠ 0
u : Set E
countable_u : u.Countable
dense_u : Dense u
key : (n : ℕ) → (Fin n → ↑u) → (Fin n → ↑𝒜₀) → ↥(Lp E p μ) := fun n d s ↦ ∑ i, indicatorConstLp p ⋯ ⋯ ↑(d i)
D : Set ↥(Lp E p μ) := {s | ∃ n d t, s = key n d t}
a : E
s : Set X
ms : MeasurableSet s
hμs✝ : μ s < ∞
hμs : μ s ≠ ∞
⊢ ∀ (ε : ℝ), 0 < ε → ∃ b ∈ D, ‖↑(simpleFunc.indicatorConst p ms ⋯ a) - b‖ < ε

Tactic:
]

Goal state after:
case refine_2.refine_1
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : IsSeparable μ
inst✝ : SeparableSpace E
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
𝒜₀ : Set (Set X) := {s | s ∈ 𝒜 ∧ μ s ≠ ∞}
h𝒜₀ : μ.MeasureDense 𝒜₀
count_𝒜₀ : 𝒜₀.Countable
p_ne_zero : p ≠ 0
u : Set E
countable_u : u.Countable
dense_u : Dense u
key : (n : ℕ) → (Fin n → ↑u) → (Fin n → ↑𝒜₀) → ↥(Lp E p μ) := fun n d s ↦ ∑ i, indicatorConstLp p ⋯ ⋯ ↑(d i)
D : Set ↥(Lp E p μ) := {s | ∃ n d t, s = key n d t}
a : E
s : Set X
ms : MeasurableSet s
hμs✝ : μ s < ∞
hμs : μ s ≠ ∞
⊢ ∀ (ε : ℝ), 0 < ε → ∃ b ∈ D, ‖↑(simpleFunc.indicatorConst p ms ⋯ a) - b‖ < ε
Module: Mathlib.MeasureTheory.Order.UpperLower

Goal state before:
pp: case pos
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs : IsUpperSet s
x : ι → ℝ
hx : x ∈ frontier s
h : x ∈ closure s
x✝ : ℝ
⊢ x ∈ frontier s

Tactic:
]

Goal state after:
case pos
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs : IsUpperSet s
x : ι → ℝ
hx : x ∈ frontier s
h : x ∈ closure s
x✝ : ℝ
⊢ x ∈ frontier s
Module: Mathlib.MeasureTheory.Order.UpperLower

Goal state before:
pp: case pos
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs : IsLowerSet s
x : ι → ℝ
hx : x ∈ frontier s
h : x ∈ closure s
x✝ : ℝ
⊢ x ∈ frontier s

Tactic:
]

Goal state after:
case pos
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs : IsLowerSet s
x : ι → ℝ
hx : x ∈ frontier s
h : x ∈ closure s
x✝ : ℝ
⊢ x ∈ frontier s
Module: Mathlib.MeasureTheory.Order.UpperLower

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs : s.OrdConnected
⊢ volume (frontier (↑(upperClosure s) ∩ ↑(lowerClosure s))) = 0

Tactic:
]

Goal state after:
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs : s.OrdConnected
⊢ volume (frontier (↑(upperClosure s) ∩ ↑(lowerClosure s))) = 0
Module: Mathlib.MeasureTheory.Order.UpperLower

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
s : Set (ι → ℝ)
inst✝ : Nonempty ι
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s
⊢ s ⊆ closure s

Tactic:
]

Goal state after:
ι : Type u_1
inst✝¹ : Fintype ι
s : Set (ι → ℝ)
inst✝ : Nonempty ι
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s
⊢ s ⊆ closure s
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝² : PseudoMetricSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : ⋃ i, U i = univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : ℕ → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction : ∀ (k : ℕ), ↑((μ k) (⋃ i, ⋃ (_ : i ≤ k), U i)) ≤ 1 - ↑(NNReal.mk ε hε')
μlim : ProbabilityMeasure 𝓧
left✝ : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
n : ℕ
b : ℝ
c : ℕ
h : ∀ (b_1 : ℕ), c ≤ b_1 → b ≤ ↑((μ (sub b_1)) (⋃ i, ⋃ (_ : i ≤ sub b_1), U i))
⊢ ε ≤ 1

Tactic:
]

Goal state after:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝² : PseudoMetricSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : ⋃ i, U i = univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : ℕ → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction : ∀ (k : ℕ), ↑((μ k) (⋃ i, ⋃ (_ : i ≤ k), U i)) ≤ 1 - ↑(NNReal.mk ε hε')
μlim : ProbabilityMeasure 𝓧
left✝ : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
n : ℕ
b : ℝ
c : ℕ
h : ∀ (b_1 : ℕ), c ≤ b_1 → b ≤ ↑((μ (sub b_1)) (⋃ i, ⋃ (_ : i ≤ sub b_1), U i))
⊢ ε ≤ 1
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case refine_2.refine_1
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : IsSeparable μ
inst✝ : SeparableSpace E
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
𝒜₀ : Set (Set X) := {s | s ∈ 𝒜 ∧ μ s ≠ ∞}
h𝒜₀ : μ.MeasureDense 𝒜₀
count_𝒜₀ : 𝒜₀.Countable
p_ne_zero : p ≠ 0
u : Set E
countable_u : u.Countable
dense_u : Dense u
key : (n : ℕ) → (Fin n → ↑u) → (Fin n → ↑𝒜₀) → ↥(Lp E p μ) := fun n d s ↦ ∑ i, indicatorConstLp p ⋯ ⋯ ↑(d i)
D : Set ↥(Lp E p μ) := {s | ∃ n d t, s = key n d t}
a : E
s : Set X
ms : MeasurableSet s
hμs✝ : μ s < ∞
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
μs_pow_nonneg : 0 ≤ μ.real s ^ (1 / p.toReal)
approx_a_pos : 0 < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
b : E
b_mem : b ∈ u
hb : ‖a - b‖ < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
t : Set X
ht : t ∈ 𝒜₀
hμt : μ t ≠ ∞
hst : ‖indicatorConstLp p ms hμs b - indicatorConstLp p ⋯ hμt b‖ < ε / 3
mt : MeasurableSet t
⊢ ‖indicatorConstLp p ms ⋯ a - indicatorConstLp p ms hμs b +
        (indicatorConstLp p ms hμs b - indicatorConstLp p mt hμt b)‖ <
    ε / 2 + ε / 2

Tactic:
]

Goal state after:
case refine_2.refine_1
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : IsSeparable μ
inst✝ : SeparableSpace E
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
𝒜₀ : Set (Set X) := {s | s ∈ 𝒜 ∧ μ s ≠ ∞}
h𝒜₀ : μ.MeasureDense 𝒜₀
count_𝒜₀ : 𝒜₀.Countable
p_ne_zero : p ≠ 0
u : Set E
countable_u : u.Countable
dense_u : Dense u
key : (n : ℕ) → (Fin n → ↑u) → (Fin n → ↑𝒜₀) → ↥(Lp E p μ) := fun n d s ↦ ∑ i, indicatorConstLp p ⋯ ⋯ ↑(d i)
D : Set ↥(Lp E p μ) := {s | ∃ n d t, s = key n d t}
a : E
s : Set X
ms : MeasurableSet s
hμs✝ : μ s < ∞
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
μs_pow_nonneg : 0 ≤ μ.real s ^ (1 / p.toReal)
approx_a_pos : 0 < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
b : E
b_mem : b ∈ u
hb : ‖a - b‖ < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
t : Set X
ht : t ∈ 𝒜₀
hμt : μ t ≠ ∞
hst : ‖indicatorConstLp p ms hμs b - indicatorConstLp p ⋯ hμt b‖ < ε / 3
mt : MeasurableSet t
⊢ ‖indicatorConstLp p ms ⋯ a - indicatorConstLp p ms hμs b +
        (indicatorConstLp p ms hμs b - indicatorConstLp p mt hμt b)‖ <
    ε / 2 + ε / 2
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case h
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝² : PseudoMetricSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : ⋃ i, U i = univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : ℕ → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction : ∀ (k : ℕ), ↑((μ k) (⋃ i, ⋃ (_ : i ≤ k), U i)) ≤ 1 - ↑(NNReal.mk ε hε')
μlim : ProbabilityMeasure 𝓧
left✝ : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
n : ℕ
b : ℝ
c : ℕ
h : ∀ (b_1 : ℕ), c ≤ b_1 → b ≤ ↑((μ (sub b_1)) (⋃ i, ⋃ (_ : i ≤ sub b_1), U i))
⊢ ↑((μ (sub c)) (⋃ i, ⋃ (_ : i ≤ sub c), U i)) ≤ ENNReal.ofReal (1 - ε)

Tactic:
]

Goal state after:
case h
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝² : PseudoMetricSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : ⋃ i, U i = univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : ℕ → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction : ∀ (k : ℕ), ↑((μ k) (⋃ i, ⋃ (_ : i ≤ k), U i)) ≤ 1 - ↑(NNReal.mk ε hε')
μlim : ProbabilityMeasure 𝓧
left✝ : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
n : ℕ
b : ℝ
c : ℕ
h : ∀ (b_1 : ℕ), c ≤ b_1 → b ≤ ↑((μ (sub b_1)) (⋃ i, ⋃ (_ : i ≤ sub b_1), U i))
⊢ ↑((μ (sub c)) (⋃ i, ⋃ (_ : i ≤ sub c), U i)) ≤ ENNReal.ofReal (1 - ε)