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

Goal state before:
pp: α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ ν : Measure α
s : Set ι
t : ι → Set α
hc : s.Countable
⊢ μ.restrict (⋃ i ∈ s, t i) = ν.restrict (⋃ i ∈ s, t i) ↔ ∀ i ∈ s, μ.restrict (t i) = ν.restrict (t i)

Tactic:
have := hc.toEncodable
  simp only [biUnion_eq_iUnion, SetCoe.forall', restrict_iUnion_congr]

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

Goal state before:
pp: case hs
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ MeasurableSet (f ⁻¹' s)

pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ Measurable f

Tactic:
· exact ⟨s, hs, rfl⟩

Goal state after:
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ Measurable f
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: case refine_1.refine_2
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
n : ℕ
x✝ : MeasurableSet (natGeneratingSequence α n)
x : α
p : ℕ → Prop
hpn : p n
hpx : ∀ (i : ℕ), x ∈ if p i then natGeneratingSequence α i else (natGeneratingSequence α i)ᶜ
⊢ x ∈ natGeneratingSequence α n

Tactic:
specialize hpx n

Goal state after:
case refine_1.refine_2
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
n : ℕ
x✝ : MeasurableSet (natGeneratingSequence α n)
x : α
p : ℕ → Prop
hpn : p n
hpx : x ∈ if p n then natGeneratingSequence α n else (natGeneratingSequence α n)ᶜ
⊢ x ∈ natGeneratingSequence α n
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
p : α → Prop
⊢ (∀ᵐ (x : α) ∂μ.restrict s, p x) ↔ ∀ᵐ (x : ↑s) ∂Measure.comap Subtype.val μ, p ↑x

Tactic:
rw [← map_comap_subtype_coe hs, (MeasurableEmbedding.subtype_coe hs).ae_map_iff]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
t : Set β
ht : 0 ∉ t
⊢ μ (s.ite (f ⁻¹' t) (0 ⁻¹' t))ᶜ = 0 ↔ μ ((f ⁻¹' t)ᶜ ∪ sᶜ) = 0

Tactic:
Set.ite,

Goal state after:
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
t : Set β
ht : 0 ∉ t
⊢ μ (f ⁻¹' t ∩ s ∪ 0 ⁻¹' t \ s)ᶜ = 0 ↔ μ ((f ⁻¹' t)ᶜ ∪ sᶜ) = 0
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: case measurable_invFun
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : CountablyGenerated α
inst✝ : SeparatesPoints α
⊢ autoParam (Measurable ⇑(Equiv.ofInjective (mapNatBool α) ⋯).symm) MeasurableEquiv.measurable_invFun._autoParam

Tactic:
apply measurable_generateFrom

Goal state after:
case measurable_invFun
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : CountablyGenerated α
inst✝ : SeparatesPoints α
⊢ ∀ t ∈ range (natGeneratingSequence α), MeasurableSet (⇑(Equiv.ofInjective (mapNatBool α) ⋯).symm ⁻¹' t)
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
t : ℕ → Set α
n : ℕ
s u : Set α
hu : u ∈ memPartition t n
ht✝ : MeasurableSet u
⊢ ↑{u} ⊆ memPartition t n

Tactic:
simp [hu]

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

Goal state before:
pp: α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
iUnion_P : ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i
P_subset_s : ∀ {i : ι} {C : Finset ι}, i ∈ C → P C ⊆ s i
mem_C : ∀ {i : ι}, i ∈ F → ∀ {C : Finset ι} {x : α}, x ∈ P C → x ∈ s i → i ∈ C
⊢ ∑ i ∈ F, (μ.restrict (s i)) t ≤ (M • μ.restrict (⋃ i, s i)) t

Tactic:
have C_subset_C {C₁ C₂} (hC₁ : C₁ ∈ Cs) {x : α} (hx : x ∈ P C₁ ∩ P C₂) : C₁ ⊆ C₂ :=
    fun i hi ↦ mem_C (mem_powerset.mp (sdiff_subset hC₁) hi) hx.2 <| P_subset_s hi hx.1

Goal state after:
α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
iUnion_P : ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i
P_subset_s : ∀ {i : ι} {C : Finset ι}, i ∈ C → P C ⊆ s i
mem_C : ∀ {i : ι}, i ∈ F → ∀ {C : Finset ι} {x : α}, x ∈ P C → x ∈ s i → i ∈ C
C_subset_C : ∀ {C₁ C₂ : Finset ι}, C₁ ∈ Cs → ∀ {x : α}, x ∈ P C₁ ∩ P C₂ → C₁ ⊆ C₂
⊢ ∑ i ∈ F, (μ.restrict (s i)) t ≤ (M • μ.restrict (⋃ i, s i)) t
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
⊢ AEMeasurable g (μ.restrict (Ioi x))

Tactic:
obtain ⟨u, hu_tendsto⟩ := exists_seq_tendsto (atTop : Filter α)

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
⊢ AEMeasurable g (μ.restrict (Ioi x))
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: case cons
M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, Measurable f) → Measurable l.prod
hl : ∀ f_1 ∈ f :: l, Measurable f_1
⊢ Measurable (f :: l).prod

Tactic:
rw [List.forall_mem_cons] at hl

Goal state after:
case cons
M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, Measurable f) → Measurable l.prod
hl : Measurable f ∧ ∀ x ∈ l, Measurable x
⊢ Measurable (f :: l).prod
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: case cons
M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
μ : Measure α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEMeasurable f μ) → AEMeasurable l.prod μ
hl : ∀ f_1 ∈ f :: l, AEMeasurable f_1 μ
⊢ AEMeasurable (f :: l).prod μ

Tactic:
rw [List.forall_mem_cons] at hl

Goal state after:
case cons
M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
μ : Measure α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEMeasurable f μ) → AEMeasurable l.prod μ
hl : AEMeasurable f μ ∧ ∀ x ∈ l, AEMeasurable x μ
⊢ AEMeasurable (f :: l).prod μ
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: γ : Type u_3
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : R1Space γ
K s : Set γ
hK : IsCompact K
hs : MeasurableSet s
hKs : K ⊆ s
⊢ ∀ i ∈ K, {y | Inseparable i y} ⊆ s

Tactic:
exact fun x hx y hy ↦ (hy.mem_measurableSet_iff hs).1 (hKs hx)

Goal state after:
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x y : α
s : Set α
hy : y ∈ s
i : α
hi : i ∈ s
⊢ edist x y ≤ edist x i + ediam s

Tactic:
calc
    edist x y ≤ edist x i + edist i y := edist_triangle _ _ _
    _ ≤ edist x i + Metric.ediam s := add_le_add le_rfl (Metric.edist_le_ediam_of_mem hi hy)

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

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
h : SecondCountableTopologyEither α β
s : Set (α × β)
hs : IsOpen[instTopologicalSpaceProd] s
hα : SecondCountableTopology α
⊢ MeasurableSet s

pp: case inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
h : SecondCountableTopologyEither α β
s : Set (α × β)
hs : IsOpen[instTopologicalSpaceProd] s
hβ : SecondCountableTopology β
⊢ MeasurableSet s

Tactic:
· let F : Set α → Set β := fun a ↦ {y | ∃ b, IsOpen b ∧ y ∈ b ∧ a ×ˢ b ⊆ s}
    have A : ∀ a, IsOpen (F a) := by
      intro a
      apply isOpen_iff_forall_mem_open.2
      rintro y ⟨b, b_open, yb, hb⟩
      exact ⟨b, fun z zb ↦ ⟨b, b_open, zb, hb⟩, b_open, yb⟩
    have : s = ⋃ a ∈ countableBasis α, a ×ˢ F a := by
      apply Subset.antisymm
      · rintro ⟨y1, y2⟩ hy
        rcases isOpen_prod_iff.1 hs y1 y2 hy with ⟨u, v, u_open, v_open, yu, yv, huv⟩
        obtain ⟨a, ha, ya, au⟩ : ∃ a ∈ countableBasis α, y1 ∈ a ∧ a ⊆ u :=
          IsTopologicalBasis.exists_subset_of_mem_open (isBasis_countableBasis α) yu u_open
        simp only [mem_iUnion, mem_prod, exists_and_left, exists_prop]
        exact ⟨a, ya, ha, v, v_open, yv, (Set.prod_mono_left au).trans huv⟩
      · rintro ⟨y1, y2⟩ hy
        simp only [mem_iUnion, mem_prod, exists_and_left, exists_prop] at hy
        rcases hy with ⟨a, ya, -, b, -, yb, hb⟩
        exact hb (mem_prod.2 ⟨ya, yb⟩)
    rw [this]
    apply MeasurableSet.biUnion (countable_countableBasis α) (fun a ha ↦ ?_)
    exact (isOpen_of_mem_countableBasis ha).measurableSet.prod (A a).measurableSet

Goal state after:
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
h : SecondCountableTopologyEither α β
s : Set (α × β)
hs : IsOpen[instTopologicalSpaceProd] s
hβ : SecondCountableTopology β
⊢ MeasurableSet s
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: α : Type u
inst✝¹ : PseudoEMetricSpace α
s : Set α
inst✝ : LocallyCompactSpace α
hs : IsCompact s
K : Set α
K_compact : IsCompact K
hK : s ⊆ interior K
δ : ℝ
δpos : 0 < δ
hδ : cthickening δ s ⊆ interior K
⊢ IsCompact (cthickening δ s)

Tactic:
exact K_compact.of_isClosed_subset isClosed_cthickening (hδ.trans interior_subset)

Goal state after:
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case he
α : Type u
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : PseudoEMetricSpace β
f : α → β
K : Set α
U : Set β
hK : IsCompact K
ho : IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] U
hf : ∀ x ∈ K, ContinuousAt f x
hKU : MapsTo f K U
⊢ ∃ ε > 0, ∃ V ∈ 𝓝ˢ ∅, thickening ε (f '' V) ⊆ U

Tactic:
use 1, by positivity, ∅, by simp, by simp

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

Goal state before:
pp: α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsLUB (range fun i ↦ f i b) (g b)
⊢ Measurable g

Tactic:
apply measurable_generateFrom

Goal state after:
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsLUB (range fun i ↦ f i b) (g b)
⊢ ∀ t ∈ range Ioi, MeasurableSet (g ⁻¹' t)
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
g : β → ℝ≥0∞
f : α →ₛ β
⊢ ∑ x ∈ Finset.image g f.range, x * μ (⇑(map g f) ⁻¹' {x}) = ∑ x ∈ f.range, g x * μ (⇑f ⁻¹' {x})

Tactic:
refine Finset.sum_image' _ fun b hb => ?_

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
g : β → ℝ≥0∞
f : α →ₛ β
b : β
hb : b ∈ f.range
⊢ g b * μ (⇑(map g f) ⁻¹' {g b}) = ∑ j ∈ f.range with g j = g b, g j * μ (⇑f ⁻¹' {j})
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case inr
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
⊢ MeasurableSet {b | BddAbove (range fun i ↦ f i b)}

Tactic:
obtain ⟨u, hu⟩ : ∃ (u : ℕ → α), Tendsto u atTop atTop := exists_seq_tendsto (atTop : Filter α)

Goal state after:
case inr
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
u : ℕ → α
hu : Tendsto u atTop atTop
⊢ MeasurableSet {b | BddAbove (range fun i ↦ f i b)}
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
s : Set ι
f : ι → δ → α
hs : s.Countable
hf : ∀ i ∈ s, Measurable (f i)
this : Encodable ↑s
⊢ Measurable fun b ↦ ⨆ i ∈ s, f i b

Tactic:
by_cases H : ∀ i, i ∈ s

Goal state after:
case pos
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
s : Set ι
f : ι → δ → α
hs : s.Countable
hf : ∀ i ∈ s, Measurable (f i)
this : Encodable ↑s
H : ∀ (i : ι), i ∈ s
⊢ Measurable fun b ↦ ⨆ i ∈ s, f i b

case neg
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
s : Set ι
f : ι → δ → α
hs : s.Countable
hf : ∀ i ∈ s, Measurable (f i)
this : Encodable ↑s
H : ¬∀ (i : ι), i ∈ s
⊢ Measurable fun b ↦ ⨆ i ∈ s, f i b
Module: Mathlib.Topology.IndicatorConstPointwise

Goal state before:
pp: case neg
ι : Type u_3
L : Filter ι
β : Type u_4
p : ι → Prop
inst✝¹ : DecidablePred p
q : Prop
inst✝ : Decidable q
a b : β
F G : Filter β
haG : {a}ᶜ ∈ G
hbF : {b}ᶜ ∈ F
haF : 𝓟 {a} ≤ F
hbG : 𝓟 {b} ≤ G
hq : ¬q
h : Tendsto (fun i ↦ if p i then a else b) L G
⊢ ∀ᶠ (i : ι) in L, p i ↔ q

Tactic:
filter_upwards [mem_map.mp (h haG)] with i hi

Goal state after:
ι : Type u_3
L : Filter ι
β : Type u_4
p : ι → Prop
inst✝¹ : DecidablePred p
q : Prop
inst✝ : Decidable q
a b : β
F G : Filter β
haG : {a}ᶜ ∈ G
hbF : {b}ᶜ ∈ F
haF : 𝓟 {a} ≤ F
hbG : 𝓟 {b} ≤ G
hq : ¬q
h : Tendsto (fun i ↦ if p i then a else b) L G
i : ι
hi : i ∈ (fun i ↦ if p i then a else b) ⁻¹' {a}ᶜ
⊢ p i ↔ q
Module: Mathlib.Topology.IndicatorConstPointwise

Goal state before:
pp: case neg
ι : Type u_3
L : Filter ι
β : Type u_4
p : ι → Prop
inst✝¹ : DecidablePred p
q : Prop
inst✝ : Decidable q
a b : β
F G : Filter β
haG : {a}ᶜ ∈ G
hbF : {b}ᶜ ∈ F
haF : 𝓟 {a} ≤ F
hbG : 𝓟 {b} ≤ G
h : Tendsto (fun i ↦ if p i then a else b) L (if q then F else G)
hq : ¬q
⊢ ∀ᶠ (i : ι) in L, p i ↔ q

Tactic:
· simp only [hq, ite_false] at h
      filter_upwards [mem_map.mp (h haG)] with i hi
      simp only [Set.preimage_compl, Set.mem_compl_iff, Set.mem_preimage, Set.mem_singleton_iff,
        ite_eq_left_iff, not_forall, exists_prop] at hi
      tauto

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

Goal state before:
pp: ⊢ borel ℝ = generateFrom (⋃ a, {Ici ↑a})

Tactic:
rw [borel_eq_generateFrom_Iio_rat, iUnion_singleton_eq_range, iUnion_singleton_eq_range]

Goal state after:
⊢ generateFrom (range fun a ↦ Iio ↑a) = generateFrom (range fun a ↦ Ici ↑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 : ∀ f_1 ∈ f :: l, AEStronglyMeasurable f_1 μ
⊢ AEStronglyMeasurable (f :: l).prod μ

Tactic:
rw [List.forall_mem_cons] at hl

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 0)
hfx0 : f x = 0
h_tendsto_norm : Tendsto (fun n ↦ ‖(hf.approx n) x‖) atTop (𝓝 0)
⊢ Tendsto (fun n ↦ min 1 (c / ‖(hf.approx n) x‖) • (hf.approx n) x) atTop (𝓝 0)

Tactic:
refine squeeze_zero_norm (fun n => ?_) h_tendsto_norm

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
h_tendsto_norm : Tendsto (fun n ↦ ‖(hf.approx n) x‖) atTop (𝓝 0)
n : ℕ
⊢ ‖min 1 (c / ‖(hf.approx n) x‖) • (hf.approx n) x‖ ≤ ‖(hf.approx n) x‖
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Markov

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
f : α → ℝ≥0∞
hf : ∀ a ∈ s, a ∈ t → f a ≤ 1
hf' : ∀ a ∈ s, a ∉ t → f a = 0
⊢ ∀ a ∈ tᶜ, s.indicator f a = 0

Tactic:
by simp_all

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

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Countable β
f : β → α → ℝ≥0∞
μ : Measure α
hμ : μ ≠ 0
hf : ∀ (b : β), Measurable (f b)
hf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ∞
h_directed : Directed (fun x1 x2 ↦ x1 ≥ x2) f
val✝ : Encodable β
h✝ : Nonempty β
inhabited_h : Inhabited β
this : ∀ (a : α), ⨅ b, f b a = ⨅ n, f (Directed.sequence f h_directed n) a
n : ℕ
⊢ ⨅ b, ∫⁻ (a : α), f b a ∂μ ≤ ∫⁻ (a : α), (f ∘ Directed.sequence f h_directed) n a ∂μ

Tactic:
· exact iInf_le (fun b => ∫⁻ a, f b a ∂μ) _

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

Goal state before:
pp: case cons
α : Type u_1
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
m : MeasurableSpace α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, StronglyMeasurable f) → StronglyMeasurable l.prod
hl : ∀ f_1 ∈ f :: l, StronglyMeasurable f_1
⊢ StronglyMeasurable (f :: l).prod

Tactic:
rw [List.forall_mem_cons] at hl

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
g φ : α → ℝ≥0∞
hφm : Measurable φ
hφ_le : φ ≤ fun a ↦ f a + g a
hφ_eq : ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), φ a ∂μ
⊢ ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), φ a - f a ∂μ ≤ ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ

Tactic:
gcongr with a

Goal state after:
case hfg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
g φ : α → ℝ≥0∞
hφm : Measurable φ
hφ_le : φ ≤ fun a ↦ f a + g a
hφ_eq : ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), φ a ∂μ
a : α
⊢ φ a - f a ≤ g a
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
E : Type u_5
m : MeasurableSpace α
f g : α → E
inst✝¹ : TopologicalSpace E
inst✝ : MetrizableSpace E
hf : StronglyMeasurable f
hg : StronglyMeasurable g
⊢ MeasurableSet {a | f a = g a}

Tactic:
by
  borelize (E × E)
  exact (hf.prodMk hg).measurable isClosed_diagonal.measurableSet

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : SFinite ν
hs_subset : s ⊆ (μ.sigmaFiniteSetWRT ν)ᶜ
hνs : ν s ≠ 0
ν' : Measure α
hν' : IsFiniteMeasure ν'
hνν' : ν ≪ ν'
right✝ : ν' ≪ ν
⊢ μ s = ∞

Tactic:
have h : ∃ s : Set α, MeasurableSet s ∧ SigmaFinite (μ.restrict s)
      ∧ (∀ t ⊆ sᶜ, ν t ≠ 0 → μ t = ∞) := by
    refine ⟨μ.sigmaFiniteSetWRT' ν', measurableSet_sigmaFiniteSetWRT',
      sigmaFinite_restrict_sigmaFiniteSetWRT' _ _,
      fun t ht_subset hνt ↦ measure_eq_top_of_subset_compl_sigmaFiniteSetWRT' ht_subset ?_⟩
    exact fun hν't ↦ hνt (hνν' hν't)

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : SFinite ν
hs_subset : s ⊆ (μ.sigmaFiniteSetWRT ν)ᶜ
hνs : ν s ≠ 0
ν' : Measure α
hν' : IsFiniteMeasure ν'
hνν' : ν ≪ ν'
right✝ : ν' ≪ ν
h : ∃ s, MeasurableSet s ∧ SigmaFinite (μ.restrict s) ∧ ∀ t ⊆ sᶜ, ν t ≠ 0 → μ t = ∞
⊢ μ s = ∞