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

Goal state before:
pp: α : Type u_5
γ : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : Nonempty γ
P : (α →ₛ γ) → Prop
const : ∀ (c : γ), P (SimpleFunc.const α c)
pcw : ∀ ⦃f g : α →ₛ γ⦄ {s : Set α} (hs : MeasurableSet s), P f → P g → P (piecewise s hs f g)
c : γ := Classical.ofNonempty
x : γ
s : Finset γ
hxs : x ∉ s
ih : ∀ (f : α →ₛ γ), range ⇑f \ {c} = ↑s → P f
f : α →ₛ γ
h : range ⇑f \ {c} = ↑(insert x s)
mx : MeasurableSet (⇑f ⁻¹' {x})
g : α →ₛ γ := piecewise (⇑f ⁻¹' {x}) mx (SimpleFunc.const α c) f
⊢ x ∉ s

Tactic:
]

Goal state after:
α : Type u_5
γ : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : Nonempty γ
P : (α →ₛ γ) → Prop
const : ∀ (c : γ), P (SimpleFunc.const α c)
pcw : ∀ ⦃f g : α →ₛ γ⦄ {s : Set α} (hs : MeasurableSet s), P f → P g → P (piecewise s hs f g)
c : γ := Classical.ofNonempty
x : γ
s : Finset γ
hxs : x ∉ s
ih : ∀ (f : α →ₛ γ), range ⇑f \ {c} = ↑s → P f
f : α →ₛ γ
h : range ⇑f \ {c} = ↑(insert x s)
mx : MeasurableSet (⇑f ⁻¹' {x})
g : α →ₛ γ := piecewise (⇑f ⁻¹' {x}) mx (SimpleFunc.const α c) f
⊢ x ∉ s
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
⊢ μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞} ∪ s ∩ f ⁻¹' Ioo 0 ∞)

pp: case hd
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
⊢ Disjoint (s ∩ f ⁻¹' {∞}) (s ∩ f ⁻¹' Ioo 0 ∞)

pp: case h
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
⊢ MeasurableSet (s ∩ f ⁻¹' Ioo 0 ∞)

Tactic:
]

Goal state after:
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
⊢ μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞} ∪ s ∩ f ⁻¹' Ioo 0 ∞)

case hd
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
⊢ Disjoint (s ∩ f ⁻¹' {∞}) (s ∩ f ⁻¹' Ioo 0 ∞)

case h
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
⊢ MeasurableSet (s ∩ f ⁻¹' Ioo 0 ∞)
Module: Mathlib.MeasureTheory.Function.SimpleFuncDense

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : PseudoEMetricSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : MeasurableSpace β
f : β → α
hf : Measurable f
s : Set α
y₀ : α
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
x : β
this : Nonempty ↑s
hx :
  f x ∈
    closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace]
      (Subtype.val '' closure[instTopologicalSpaceSubtype] (range (denseSeq ↑s)))
⊢ Tendsto (fun n ↦ (approxOn f hf s y₀ h₀ n) x) atTop (𝓝 (f x))

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : PseudoEMetricSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : MeasurableSpace β
f : β → α
hf : Measurable f
s : Set α
y₀ : α
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
x : β
this : Nonempty ↑s
hx :
  f x ∈
    closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace]
      (Subtype.val '' closure[instTopologicalSpaceSubtype] (range (denseSeq ↑s)))
⊢ Tendsto (fun n ↦ (approxOn f hf s y₀ h₀ n) x) atTop (𝓝 (f x))
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
⊢ μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞} ∪ s ∩ f ⁻¹' Ioo 0 ∞)

Tactic:
← inter_union_distrib_left

Goal state after:
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
⊢ μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ (f ⁻¹' {∞} ∪ f ⁻¹' Ioo 0 ∞))
Module: Mathlib.MeasureTheory.Function.SimpleFuncDense

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : PseudoEMetricSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : MeasurableSpace β
f : β → α
hf : Measurable f
s : Set α
y₀ : α
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
x : β
n : ℕ
⊢ edist y₀ (f x) + edist y₀ (f x) ≤ edist y₀ (f x) + edist y₀ (f x)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
⊢ μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ (f ⁻¹' {∞} ∪ f ⁻¹' Ioo 0 ∞))

Tactic:
]

Goal state after:
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
⊢ μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ (f ⁻¹' {∞} ∪ f ⁻¹' Ioo 0 ∞))
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
motive : (α → ℝ≥0∞) → Prop
indicator : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → ℝ≥0∞⦄,
    Disjoint (support f) (support g) → Measurable f → Measurable g → motive f → motive g → motive (f + g)
iSup :
  ∀ ⦃f : ℕ → α → ℝ≥0∞⦄,
    (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), motive (f n)) → motive fun x ↦ ⨆ n, f n x
f : α → ℝ≥0∞
hf : Measurable f
x✝ : α
⊢ f x✝ = f x✝

Tactic:
]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
motive : (α → ℝ≥0∞) → Prop
indicator : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → ℝ≥0∞⦄,
    Disjoint (support f) (support g) → Measurable f → Measurable g → motive f → motive g → motive (f + g)
iSup :
  ∀ ⦃f : ℕ → α → ℝ≥0∞⦄,
    (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), motive (f n)) → motive fun x ↦ ⨆ n, f n x
f : α → ℝ≥0∞
hf : Measurable f
c : ℝ≥0∞
s : Set α
hs : MeasurableSet s
m n : ℕ
hmn : m ≤ n
a : α
⊢ (s ∩ spanningSets μ n).indicator (fun x ↦ c) a ≤ (s ∩ spanningSets μ n).indicator (fun x ↦ c) a

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
B : μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞}) + μ (s ∩ f ⁻¹' Ioo 0 ∞)
⊢ μ (⋃ i, s ∩ f ⁻¹' Ico (↑t ^ i) (↑t ^ (i + 1))) = μ (⋃ i, s ∩ f ⁻¹' Ico (↑t ^ i) (↑t ^ (i + 1)))

pp: case hn
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
B : μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞}) + μ (s ∩ f ⁻¹' Ioo 0 ∞)
⊢ Pairwise (Function.onFun Disjoint fun n ↦ s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))

pp: case h
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
B : μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞}) + μ (s ∩ f ⁻¹' Ioo 0 ∞)
⊢ ∀ (i : ℤ), MeasurableSet (s ∩ f ⁻¹' Ico (↑t ^ i) (↑t ^ (i + 1)))

Tactic:
]

Goal state after:
case hn
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
B : μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞}) + μ (s ∩ f ⁻¹' Ioo 0 ∞)
⊢ Pairwise (Function.onFun Disjoint fun n ↦ s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))

case h
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
B : μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞}) + μ (s ∩ f ⁻¹' Ioo 0 ∞)
⊢ ∀ (i : ℤ), MeasurableSet (s ∩ f ⁻¹' Ico (↑t ^ i) (↑t ^ (i + 1)))
Module: Mathlib.MeasureTheory.Function.SimpleFuncDense

Goal state before:
pp: case hnhds
X : Type u_3
Y : Type u_4
α : Type u_5
inst✝⁷ : Zero α
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : MeasurableSpace Y
inst✝² : OpensMeasurableSpace X
inst✝¹ : OpensMeasurableSpace Y
inst✝ : PseudoMetricSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h'f : HasCompactSupport f
ε : ℝ
hε : 0 < ε
K : Set (X × Y)
hK : IsCompact K
⊢ ∀ x ∈ K, ∃ t ∈ 𝓝[K] x, ∃ g s, MeasurableSet s ∧ t ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε

Tactic:
rintro ⟨x, y⟩ -
      obtain ⟨u, v, hu, xu, hv, yv, huv⟩ : ∃ u v, IsOpen u ∧ x ∈ u ∧ IsOpen v ∧ y ∈ v ∧
        u ×ˢ v ⊆ {z | dist (f z) (f (x, y)) < ε} :=
          mem_nhds_prod_iff'.1 <| Metric.continuousAt_iff'.1 hf.continuousAt ε hε
      refine ⟨u ×ˢ v, nhdsWithin_le_nhds <| (hu.prod hv).mem_nhds (mk_mem_prod xu yv), ?_⟩
      exact ⟨SimpleFunc.const _ (f (x, y)), u ×ˢ v, hu.measurableSet.prod hv.measurableSet,
        Subset.rfl, fun z hz ↦ huv hz⟩

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

Goal state before:
pp: X : Type u_3
Y : Type u_4
α : Type u_5
inst✝⁷ : Zero α
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : MeasurableSpace Y
inst✝² : OpensMeasurableSpace X
inst✝¹ : OpensMeasurableSpace Y
inst✝ : PseudoMetricSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h'f : HasCompactSupport f
ε : ℝ
hε : 0 < ε
⊢ ∃ g, ∀ (x : X × Y), dist (f x) (g x) < ε

Tactic:
have M : ∀ (K : Set (X × Y)), IsCompact K →
      ∃ (g : SimpleFunc (X × Y) α), ∃ (s : Set (X × Y)), MeasurableSet s ∧ K ⊆ s ∧
      ∀ x ∈ s, dist (f x) (g x) < ε := by
    intro K hK
    apply IsCompact.induction_on
      (p := fun t ↦ ∃ (g : SimpleFunc (X × Y) α), ∃ (s : Set (X × Y)), MeasurableSet s ∧ t ⊆ s ∧
        ∀ x ∈ s, dist (f x) (g x) < ε) hK
    · exact ⟨0, ∅, by simp⟩
    · intro t t' htt' ⟨g, s, s_meas, ts, hg⟩
      exact ⟨g, s, s_meas, htt'.trans ts, hg⟩
    · intro t t' ⟨g, s, s_meas, ts, hg⟩ ⟨g', s', s'_meas, t's', hg'⟩
      refine ⟨g.piecewise s s_meas g', s ∪ s', s_meas.union s'_meas,
        union_subset_union ts t's', fun p hp ↦ ?_⟩
      by_cases H : p ∈ s
      · simpa [H, SimpleFunc.piecewise_apply] using hg p H
      · simp only [SimpleFunc.piecewise_apply, H, ite_false]
        apply hg'
        simpa [H] using (mem_union _ _ _).1 hp
    · rintro ⟨x, y⟩ -
      obtain ⟨u, v, hu, xu, hv, yv, huv⟩ : ∃ u v, IsOpen u ∧ x ∈ u ∧ IsOpen v ∧ y ∈ v ∧
        u ×ˢ v ⊆ {z | dist (f z) (f (x, y)) < ε} :=
          mem_nhds_prod_iff'.1 <| Metric.continuousAt_iff'.1 hf.continuousAt ε hε
      refine ⟨u ×ˢ v, nhdsWithin_le_nhds <| (hu.prod hv).mem_nhds (mk_mem_prod xu yv), ?_⟩
      exact ⟨SimpleFunc.const _ (f (x, y)), u ×ˢ v, hu.measurableSet.prod hv.measurableSet,
        Subset.rfl, fun z hz ↦ huv hz⟩

Goal state after:
X : Type u_3
Y : Type u_4
α : Type u_5
inst✝⁷ : Zero α
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : MeasurableSpace Y
inst✝² : OpensMeasurableSpace X
inst✝¹ : OpensMeasurableSpace Y
inst✝ : PseudoMetricSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h'f : HasCompactSupport f
ε : ℝ
hε : 0 < ε
M : ∀ (K : Set (X × Y)), IsCompact K → ∃ g s, MeasurableSet s ∧ K ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε
⊢ ∃ g, ∀ (x : X × Y), dist (f x) (g x) < ε
Module: Mathlib.MeasureTheory.Function.SimpleFuncDense

Goal state before:
pp: X : Type u_3
Y : Type u_4
α : Type u_5
inst✝⁷ : Zero α
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : MeasurableSpace Y
inst✝² : OpensMeasurableSpace X
inst✝¹ : OpensMeasurableSpace Y
inst✝ : PseudoMetricSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h'f : HasCompactSupport f
ε : ℝ
hε : 0 < ε
M : ∀ (K : Set (X × Y)), IsCompact K → ∃ g s, MeasurableSet s ∧ K ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε
g : SimpleFunc (X × Y) α
s : Set (X × Y)
s_meas : MeasurableSet s
fs : tsupport f ⊆ s
hg : ∀ x ∈ s, dist (f x) (g x) < ε
p : X × Y
H : p ∈ support f
⊢ p ∈ s

Tactic:
]

Goal state after:
X : Type u_3
Y : Type u_4
α : Type u_5
inst✝⁷ : Zero α
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : MeasurableSpace Y
inst✝² : OpensMeasurableSpace X
inst✝¹ : OpensMeasurableSpace Y
inst✝ : PseudoMetricSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h'f : HasCompactSupport f
ε : ℝ
hε : 0 < ε
M : ∀ (K : Set (X × Y)), IsCompact K → ∃ g s, MeasurableSet s ∧ K ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε
g : SimpleFunc (X × Y) α
s : Set (X × Y)
s_meas : MeasurableSet s
fs : tsupport f ⊆ s
hg : ∀ x ∈ s, dist (f x) (g x) < ε
p : X × Y
H : p ∈ support f
⊢ p ∈ s
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
B : μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞}) + μ (s ∩ f ⁻¹' Ioo 0 ∞)
C : μ (s ∩ f ⁻¹' Ioo 0 ∞) = ∑' (n : ℤ), μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
⊢ μ (s ∩ f ⁻¹' {0}) + (μ (s ∩ f ⁻¹' {∞}) + ∑' (n : ℤ), μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))) =
    μ (s ∩ f ⁻¹' {0}) + (μ (s ∩ f ⁻¹' {∞}) + ∑' (n : ℤ), μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))))

Tactic:
]

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

Goal state before:
pp: X : Type u_3
Y : Type u_4
α : Type u_5
inst✝¹⁰ : Zero α
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : OpensMeasurableSpace X
inst✝⁴ : OpensMeasurableSpace Y
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetrizableSpace α
inst✝¹ : MeasurableSpace α
inst✝ : BorelSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, inst✝³] f
h'f : HasCompactSupport f
this : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
g : ℕ → SimpleFunc (X × Y) α
hg : ∀ (n : ℕ) (x : X × Y), dist (f x) ((g n) x) < u n
x : X × Y
⊢ Tendsto (fun b ↦ dist ((g b) x) (f x)) atTop (𝓝 0)

Tactic:
]

Goal state after:
X : Type u_3
Y : Type u_4
α : Type u_5
inst✝¹⁰ : Zero α
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : OpensMeasurableSpace X
inst✝⁴ : OpensMeasurableSpace Y
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetrizableSpace α
inst✝¹ : MeasurableSpace α
inst✝ : BorelSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, inst✝³] f
h'f : HasCompactSupport f
this : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
g : ℕ → SimpleFunc (X × Y) α
hg : ∀ (n : ℕ) (x : X × Y), dist (f x) ((g n) x) < u n
x : X × Y
⊢ Tendsto (fun b ↦ dist ((g b) x) (f x)) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Function.SimpleFuncDense

Goal state before:
pp: X : Type u_3
Y : Type u_4
α : Type u_5
inst✝¹⁰ : Zero α
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : OpensMeasurableSpace X
inst✝⁴ : OpensMeasurableSpace Y
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetrizableSpace α
inst✝¹ : MeasurableSpace α
inst✝ : BorelSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, inst✝³] f
h'f : HasCompactSupport f
this : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
g : ℕ → SimpleFunc (X × Y) α
hg : ∀ (n : ℕ) (x : X × Y), dist (f x) ((g n) x) < u n
x : X × Y
n : ℕ
⊢ dist (f x) ((g n) x) ≤ u n

Tactic:
]

Goal state after:
X : Type u_3
Y : Type u_4
α : Type u_5
inst✝¹⁰ : Zero α
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : OpensMeasurableSpace X
inst✝⁴ : OpensMeasurableSpace Y
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetrizableSpace α
inst✝¹ : MeasurableSpace α
inst✝ : BorelSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, inst✝³] f
h'f : HasCompactSupport f
this : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
g : ℕ → SimpleFunc (X × Y) α
hg : ∀ (n : ℕ) (x : X × Y), dist (f x) ((g n) x) < u n
x : X × Y
n : ℕ
⊢ dist (f x) ((g n) x) ≤ u n
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
f : α →ₛ ℝ≥0∞
μ : Measure α
⊢ ⨆ g, ⨆ (_ : ⇑g ≤ fun a ↦ f a), g.lintegral μ = f.lintegral μ

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
f : α →ₛ ℝ≥0∞
μ : Measure α
⊢ ⨆ g, ⨆ (_ : ⇑g ≤ fun a ↦ f a), g.lintegral μ = f.lintegral μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
hμν : μ ≤ ν
f g : α → ℝ≥0∞
hfg : f ≤ g
⊢ ⨆ g, ⨆ (_ : ⇑g ≤ fun a ↦ f a), g.lintegral μ ≤ ⨆ g_1, ⨆ (_ : ⇑g_1 ≤ fun a ↦ g a), g_1.lintegral ν

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
hμν : μ ≤ ν
f g : α → ℝ≥0∞
hfg : f ≤ g
⊢ ⨆ g, ⨆ (_ : ⇑g ≤ fun a ↦ f a), g.lintegral μ ≤ ⨆ g_1, ⨆ (_ : ⇑g_1 ≤ fun a ↦ g a), g_1.lintegral ν
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
⊢ ⨆ g, ⨆ (_ : ⇑g ≤ fun a ↦ f a), g.lintegral μ ≤ ⨆ g, ⨆ (_ : Measurable g), ⨆ (_ : g ≤ f), ∫⁻ (a : α), g a ∂μ

Tactic:
]

Goal state after:
case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
⊢ ⨆ g, ⨆ (_ : ⇑g ≤ fun a ↦ f a), g.lintegral μ ≤ ⨆ g, ⨆ (_ : Measurable g), ⨆ (_ : g ≤ f), ∫⁻ (a : α), g a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
⊢ ∫⁻ (x : α), c ∂μ = ∫⁻ (a : α), Function.const α c a ∂μ

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
⊢ ∫⁻ (x : α), c ∂μ = ∫⁻ (a : α), Function.const α c a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
⊢ μ univ = μ univ

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
c : ℝ≥0∞
⊢ c * μ s = c * μ s

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
⊢ μ s = μ s

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
m₀ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetrizableSpace α
inst✝ : OpensMeasurableSpace α
s : Set α
h1 : IsSeparable s
h2 : μ sᶜ = 0
a✝ : Nontrivial α
a : α
h : StronglyMeasurable Subtype.val
this : ((closure[inst✝²] s).piecewise id fun x ↦ a) = Function.extend Subtype.val Subtype.val fun x ↦ a
⊢ StronglyMeasurable (Function.extend Subtype.val Subtype.val fun x ↦ a)

Tactic:
]

Goal state after:
α : Type u_1
m₀ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetrizableSpace α
inst✝ : OpensMeasurableSpace α
s : Set α
h1 : IsSeparable s
h2 : μ sᶜ = 0
a✝ : Nontrivial α
a : α
h : StronglyMeasurable Subtype.val
this : ((closure[inst✝²] s).piecewise id fun x ↦ a) = Function.extend Subtype.val Subtype.val fun x ↦ a
⊢ StronglyMeasurable (Function.extend Subtype.val Subtype.val fun x ↦ a)
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: case cons
α : Type u_1
m₀ : MeasurableSpace α
μ : Measure α
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEStronglyMeasurable f μ) → AEStronglyMeasurable l.prod μ
hl : AEStronglyMeasurable f μ ∧ ∀ x ∈ l, AEStronglyMeasurable x μ
⊢ AEStronglyMeasurable (f :: l).prod μ

Tactic:
]

Goal state after:
case cons
α : Type u_1
m₀ : MeasurableSpace α
μ : Measure α
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEStronglyMeasurable f μ) → AEStronglyMeasurable l.prod μ
hl : AEStronglyMeasurable f μ ∧ ∀ x ∈ l, AEStronglyMeasurable x μ
⊢ AEStronglyMeasurable (f :: l).prod μ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: case cons
α : Type u_1
m₀ : MeasurableSpace α
μ : Measure α
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEStronglyMeasurable f μ) → AEStronglyMeasurable l.prod μ
hl : AEStronglyMeasurable f μ ∧ ∀ x ∈ l, AEStronglyMeasurable x μ
⊢ AEStronglyMeasurable (f * l.prod) μ

Tactic:
]

Goal state after:
case cons
α : Type u_1
m₀ : MeasurableSpace α
μ : Measure α
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEStronglyMeasurable f μ) → AEStronglyMeasurable l.prod μ
hl : AEStronglyMeasurable f μ ∧ ∀ x ∈ l, AEStronglyMeasurable x μ
⊢ AEStronglyMeasurable (f * l.prod) μ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case pos
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
hf : StronglyMeasurable f
c : ℝ
x : α
hfx : ‖f x‖ ≤ c
h_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 (f x))
hfx0 : f x = 0
⊢ Tendsto (fun n ↦ min 1 (c / ‖(hf.approx n) x‖) • (hf.approx n) x) atTop (𝓝 (f x))

Tactic:
]

Goal state after:
case pos
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
hf : StronglyMeasurable f
c : ℝ
x : α
hfx : ‖f x‖ ≤ c
h_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 (f x))
hfx0 : f x = 0
⊢ Tendsto (fun n ↦ min 1 (c / ‖(hf.approx n) x‖) • (hf.approx n) x) atTop (𝓝 (f x))
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
f : α → ℝ≥0∞
μ : Measure α
⊢ ⨆ g, ⨆ (_ : ⇑g ≤ fun a ↦ f a), g.lintegral μ =
    ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral μ

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
f : α → ℝ≥0∞
μ : Measure α
⊢ ⨆ g, ⨆ (_ : ⇑g ≤ fun a ↦ f a), g.lintegral μ =
    ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral μ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case pos
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
hf : StronglyMeasurable f
c : ℝ
x : α
hfx : ‖f x‖ ≤ c
h_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 0)
hfx0 : f x = 0
⊢ Tendsto (fun n ↦ min 1 (c / ‖(hf.approx n) x‖) • (hf.approx n) x) atTop (𝓝 0)

Tactic:
]

Goal state after:
case pos
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
hf : StronglyMeasurable f
c : ℝ
x : α
hfx : ‖f x‖ ≤ c
h_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 0)
hfx0 : f x = 0
⊢ Tendsto (fun n ↦ min 1 (c / ‖(hf.approx n) x‖) • (hf.approx n) x) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f : α → β
inst✝ : Zero β
s : Set α
hs : NullMeasurableSet s μ
⊢ AEStronglyMeasurable f (μ.restrict s) ↔ AEStronglyMeasurable f (μ.restrict s)

Tactic:
]

Goal state after: