mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : s i < 0
hn : ¬∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0

Tactic:
by_cases h : s ≤[i] 0
  · exact ⟨i, hi₁, Set.Subset.refl _, h, hi₂⟩
  push Not at hn
  set k := Nat.find hn
  have hk₂ : s ≤[i \ ⋃ l < k, restrictNonposSeq s i l] 0 := Nat.find_spec hn
  have hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), restrictNonposSeq s i l) :=
    MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  refine ⟨i \ ⋃ l < k, restrictNonposSeq s i l, hi₁.diff hmeas, Set.sdiff_subset, hk₂, ?_⟩
  rw [of_sdiff hmeas hi₁, s.of_disjoint_iUnion]
  · have h₁ : ∀ l < k, 0 ≤ s (restrictNonposSeq s i l) := by
      intro l hl
      refine le_of_lt (measure_of_restrictNonposSeq h _ ?_)
      refine mt (restrict_le_zero_subset _ (hi₁.diff ?_) (Set.Subset.refl _)) (Nat.find_min hn hl)
      exact
        MeasurableSet.iUnion fun _ =>
          MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
    suffices 0 ≤ ∑' l : ℕ, s (⋃ _ : l < k, restrictNonposSeq s i l) by
      rw [sub_neg]
      exact lt_of_lt_of_le hi₂ this
    refine tsum_nonneg ?_
    intro l; by_cases h : l < k
    · convert! h₁ _ h
      ext x
      rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
      exact fun _ => h
    · convert! le_of_eq s.empty.symm
      ext; simp only [exists_prop, Set.mem_empty_iff_false, Set.mem_iUnion, not_and, iff_false]
      exact fun h' => False.elim (h h')
  · intro; exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  · intro a b hab
    refine Set.disjoint_iUnion_left.mpr fun _ => ?_
    refine Set.disjoint_iUnion_right.mpr fun _ => ?_
    exact restrictNonposSeq_disjoint hab
  · apply Set.iUnion_subset
    intro a x
    simp only [and_imp, exists_prop, Set.mem_iUnion]
    intro _ hx
    exact restrictNonposSeq_subset _ hx

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
g : α → β
hp : 1 ≤ p
hp_ne_top : p ≠ ∞
hg : MemLp g p μ
ε : ℝ
⊢ 0 < ε →
    ∃ δ,
      ∃ (_ : 0 < δ),
        ∀ (i : ι) (s : Set α),
          MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator ((fun x ↦ g) i)) p μ ≤ ENNReal.ofReal ε

Tactic:
hε

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
g : α → β
hp : 1 ≤ p
hp_ne_top : p ≠ ∞
hg : MemLp g p μ
ε : ℝ
hε : 0 < ε
⊢ ∃ δ,
    ∃ (_ : 0 < δ),
      ∀ (i : ι) (s : Set α),
        MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator ((fun x ↦ g) i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Subsingleton ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), MemLp (f i) p μ
ε : ℝ
⊢ 0 < ε →
    ∃ δ,
      ∃ (_ : 0 < δ),
        ∀ (i : ι) (s : Set α),
          MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
hε

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Subsingleton ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), MemLp (f i) p μ
ε : ℝ
hε : 0 < ε
⊢ ∃ δ,
    ∃ (_ : 0 < δ),
      ∀ (i : ι) (s : Set α),
        MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case succ
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp_one : 1 ≤ p
hp_top : p ≠ ∞
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), MemLp (f i) p μ) → UnifIntegrable f p μ
f : Fin (n + 1) → α → β
hfLp : ∀ (i : Fin (n + 1)), MemLp (f i) p μ
ε : ℝ
⊢ 0 < ε →
    ∃ δ,
      ∃ (_ : 0 < δ),
        ∀ (i : Fin (n + 1)) (s : Set α),
          MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
hε

Goal state after:
case succ
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp_one : 1 ≤ p
hp_top : p ≠ ∞
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), MemLp (f i) p μ) → UnifIntegrable f p μ
f : Fin (n + 1) → α → β
hfLp : ∀ (i : Fin (n + 1)), MemLp (f i) p μ
ε : ℝ
hε : 0 < ε
⊢ ∃ δ,
    ∃ (_ : 0 < δ),
      ∀ (i : Fin (n + 1)) (s : Set α),
        MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), MemLp (f i) p μ
n : ℕ
hn : Nonempty (ι ≃ Fin n)
ε : ℝ
⊢ 0 < ε →
    ∃ δ,
      ∃ (_ : 0 < δ),
        ∀ (i : ι) (s : Set α),
          MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
hε

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), MemLp (f i) p μ
n : ℕ
hn : Nonempty (ι ≃ Fin n)
ε : ℝ
hε : 0 < ε
⊢ ∃ δ,
    ∃ (_ : 0 < δ),
      ∀ (i : ι) (s : Set α),
        MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
⊢ ε > 0 → ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε

Tactic:
hε

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε = ∞
n : ℕ
x✝ : n ≥ 0
⊢ eLpNorm (f n - g) p μ ≤ ε

Tactic:
by simp [h]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
N : Type u_5
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
M : Type u_6
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : IsTopologicalAddGroup M
v : VectorMeasure α M
w : VectorMeasure α N
h : v ⟂ᵥ w
⊢ -v ⟂ᵥ w

Tactic:
obtain ⟨u, hmu, hu₁, hu₂⟩ := h
  refine ⟨u, hmu, fun s hs => ?_, hu₂⟩
  rw [neg_apply v s, neg_eq_zero]
  exact hu₁ s hs

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
f : α → E
p : ℝ≥0∞
μ : Measure α
hf : ¬AEStronglyMeasurable f μ
⊢ lpNorm (-f) p μ = lpNorm f p μ

Tactic:
rw [lpNorm_of_not_aestronglyMeasurable hf,
      lpNorm_of_not_aestronglyMeasurable fun h ↦ hf <| by simpa using h.neg]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hf_tendsto : Tendsto (fun n ↦ eLpNorm (f n) p μ) atTop (𝓝 0)
ε : ℝ
⊢ 0 < ε →
    ∃ δ,
      ∃ (_ : 0 < δ),
        ∀ (i : ℕ) (s : Set α),
          MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
hε

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hf_tendsto : Tendsto (fun n ↦ eLpNorm (f n) p μ) atTop (𝓝 0)
ε : ℝ
hε : 0 < ε
⊢ ∃ δ,
    ∃ (_ : 0 < δ),
      ∀ (i : ℕ) (s : Set α),
        MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case neg
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
⊢ 0 < ε →
    ∃ δ,
      ∃ (_ : 0 < δ),
        ∀ (i : ι) (s : Set α),
          MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
hε

Goal state after:
case neg
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
⊢ ∃ δ,
    ∃ (_ : 0 < δ),
      ∀ (i : ι) (s : Set α),
        MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
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 + (μ.withDensityᵥ (c.rnDeriv μ)) i).re = { re := (re c) i, im := (im c) i }.re

Tactic:
Complex.add_re,

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 (c.rnDeriv μ)) i).re = { re := (re c) i, im := (im c) i }.re
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
⊢ 0 < ε →
    ∃ δ,
      ∃ (_ : 0 < δ),
        ∀ (i : ↑{g | ∃ ni, TendstoInMeasure μ (fn ∘ ni) u g}) (s : Set α),
          MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator ((fun f ↦ ↑f) i)) p μ ≤ ENNReal.ofReal ε

Tactic:
hε

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
hε : 0 < ε
⊢ ∃ δ,
    ∃ (_ : 0 < δ),
      ∀ (i : ↑{g | ∃ ni, TendstoInMeasure μ (fn ∘ ni) u g}) (s : Set α),
        MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator ((fun f ↦ ↑f) i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
⊢ 0 < ε →
    ∃ δ,
      ∃ (_ : 0 < δ),
        ∀ (i : ↑{g | ∃ ni, ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (ni n) x) u (𝓝 (g x))}) (s : Set α),
          MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator ((fun f ↦ ↑f) i)) p μ ≤ ENNReal.ofReal ε

Tactic:
hε

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
hε : 0 < ε
⊢ ∃ δ,
    ∃ (_ : 0 < δ),
      ∀ (i : ↑{g | ∃ ni, ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (ni n) x) u (𝓝 (g x))}) (s : Set α),
        MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator ((fun f ↦ ↑f) i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
m : Measure α
inst✝ : IsProbabilityMeasure m
f : α → Measure β
hf₀ : AEMeasurable f m
hf₁ : ∀ᵐ (μ : α) ∂m, IsProbabilityMeasure (f μ)
⊢ IsProbabilityMeasure (m.bind f)

Tactic:
simp only [isProbabilityMeasure_iff, MeasurableSet.univ, Measure.bind_apply _ hf₀]
  simp_rw [isProbabilityMeasure_iff] at hf₁
  exact lintegral_eq_const hf₁

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

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
inst✝² : TopologicalSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : IsFiniteMeasure μ
h : μ.InnerRegularWRT (fun s ↦ IsCompact s ∧ IsClosed[inst✝²] s) MeasurableSet
ε : ENNReal
⊢ 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ {μ}, μ_1 Kᶜ ≤ ε

Tactic:
hε

Goal state after:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
inst✝² : TopologicalSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : IsFiniteMeasure μ
h : μ.InnerRegularWRT (fun s ↦ IsCompact s ∧ IsClosed[inst✝²] s) MeasurableSet
ε : ENNReal
hε : 0 < ε
⊢ ∃ K, IsCompact K ∧ ∀ μ_1 ∈ {μ}, μ_1 Kᶜ ≤ ε
Module: Mathlib.MeasureTheory.Measure.Tight

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
S : Set (Measure 𝓧)
inst✝⁴ : TopologicalSpace 𝓧
inst✝³ : TopologicalSpace 𝓨
inst✝² : MeasurableSpace 𝓨
inst✝¹ : OpensMeasurableSpace 𝓨
inst✝ : T2Space 𝓨
hS : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ ∈ S, μ Kᶜ ≤ ε
f : 𝓧 → 𝓨
hf : Continuous[inst✝⁴, inst✝³] f
ε : ENNReal
⊢ 0 < ε → ∃ K, IsCompact K ∧ ∀ a ∈ S, (Measure.map f a) Kᶜ ≤ ε

Tactic:
hε

Goal state after:
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
S : Set (Measure 𝓧)
inst✝⁴ : TopologicalSpace 𝓧
inst✝³ : TopologicalSpace 𝓨
inst✝² : MeasurableSpace 𝓨
inst✝¹ : OpensMeasurableSpace 𝓨
inst✝ : T2Space 𝓨
hS : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ ∈ S, μ Kᶜ ≤ ε
f : 𝓧 → 𝓨
hf : Continuous[inst✝⁴, inst✝³] f
ε : ENNReal
hε : 0 < ε
⊢ ∃ K, IsCompact K ∧ ∀ a ∈ S, (Measure.map f a) Kᶜ ≤ ε
Module: Mathlib.MeasureTheory.Measure.Tight

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
inst✝¹ : TopologicalSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
inst✝ : TopologicalSpace 𝓨
μ : Set (Measure (𝓧 × 𝓨))
hμ₁ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.fst '' μ, μ_1 Kᶜ ≤ ε
hμ₂ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.snd '' μ, μ_1 Kᶜ ≤ ε
ε : ENNReal
⊢ 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ μ, μ_1 Kᶜ ≤ ε

Tactic:
hε

Goal state after:
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
inst✝¹ : TopologicalSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
inst✝ : TopologicalSpace 𝓨
μ : Set (Measure (𝓧 × 𝓨))
hμ₁ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.fst '' μ, μ_1 Kᶜ ≤ ε
hμ₂ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.snd '' μ, μ_1 Kᶜ ≤ ε
ε : ENNReal
hε : 0 < ε
⊢ ∃ K, IsCompact K ∧ ∀ μ_1 ∈ μ, μ_1 Kᶜ ≤ ε
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
p : FiniteMeasure Ω × FiniteMeasure Ω
g : Ω →ᵇ ℝ≥0
⊢ Tendsto (fun i ↦ ∫⁻ (x : Ω), ↑(g x) ∂↑(i.1 + i.2)) (𝓝 p) (𝓝 (∫⁻ (x : Ω), ↑(g x) ∂↑((fun p ↦ p.1 + p.2) p)))

Tactic:
have A : Tendsto (fun (i : FiniteMeasure Ω × FiniteMeasure Ω) ↦ ∫⁻ x, g x ∂i.1) (𝓝 p)
      (𝓝 (∫⁻ x, g x ∂p.1)) := by
    rw [nhds_prod_eq]
    exact (tendsto_iff_forall_lintegral_tendsto.1 tendsto_id g).comp tendsto_fst

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
p : FiniteMeasure Ω × FiniteMeasure Ω
g : Ω →ᵇ ℝ≥0
A : Tendsto (fun i ↦ ∫⁻ (x : Ω), ↑(g x) ∂↑i.1) (𝓝 p) (𝓝 (∫⁻ (x : Ω), ↑(g x) ∂↑p.1))
⊢ Tendsto (fun i ↦ ∫⁻ (x : Ω), ↑(g x) ∂↑(i.1 + i.2)) (𝓝 p) (𝓝 (∫⁻ (x : Ω), ↑(g x) ∂↑((fun p ↦ p.1 + p.2) p)))
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
β : ι → Type u_10
mβ : (i : ι) → MeasurableSpace (β i)
f g : (i : ι) → Ω → β i
h : ∀ (i : ι), f i =ᵐ[μ] g i
⊢ ∀ (i : ι), ∀ᵐ (a : Unit) ∂Measure.dirac (), f i =ᵐ[(Kernel.const Unit μ) a] g i

Tactic:
by simp [h]

Goal state after:
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : Fintype ι
β : ι → Type u_11
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf : ∀ (i : ι), AEMeasurable (f i) μ
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → μ (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, μ (f i ⁻¹' sets i)
this : IsProbabilityMeasure μ
h₀ :
  ∀ {s : (i : ι) → Set (β i)},
    (∀ (i : ι), MeasurableSet (s i)) →
      ∏ i, μ (f i ⁻¹' s i) = ∏ i, (Measure.map (f i) μ) (s i) ∧
        μ (⋂ i, f i ⁻¹' s i) = (Measure.map (fun ω i ↦ f i ω) μ) (univ.pi s)
h' : (i : ι) → Set (β i)
hm : ∀ (i : ι), MeasurableSet (h' i)
⊢ μ (⋂ i, f i ⁻¹' h' i) = ∏ i, μ (f i ⁻¹' h' i)

Tactic:
simpa [hm] using h Finset.univ (sets := h')

Goal state after:
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : Fintype ι
β : ι → Type u_11
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
inst✝ : IsProbabilityMeasure μ
hf : ∀ (i : ι), AEMeasurable (f i) μ
⊢ iIndepFun f μ ↔ Measure.map (fun ω i ↦ f i ω) μ = Measure.pi fun i ↦ Measure.map (f i) μ

Tactic:
refine ⟨iIndepFun.map_fun_eq_pi_map hf, ?_⟩

Goal state after:
Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : Fintype ι
β : ι → Type u_11
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
inst✝ : IsProbabilityMeasure μ
hf : ∀ (i : ι), AEMeasurable (f i) μ
⊢ (Measure.map (fun ω i ↦ f i ω) μ = Measure.pi fun i ↦ Measure.map (f i) μ) → iIndepFun f μ
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inl
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set (Set Ω)
S T : Set ι
h_indep : iIndepSets s κ μ
hST : Disjoint S T
t1 t2 : Set Ω
p1 : Finset ι
hp1 : ↑p1 ⊆ S
f1 : ι → Set Ω
ht1_m : ∀ x ∈ p1, f1 x ∈ s x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
p2 : Finset ι
hp2 : ↑p2 ⊆ T
f2 : ι → Set Ω
ht2_m : ∀ x ∈ p2, f2 x ∈ s x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
g : ι → Set Ω := fun i ↦ (if i ∈ p1 then f1 i else univ) ∩ if i ∈ p2 then f2 i else univ
i : ι
hi1 : i ∈ p1
hi2 : i ∉ p2
⊢ (f1 i ∩ if i ∈ p2 then f2 i else univ) ∈ s i

Tactic:
if_neg hi2,

Goal state after:
case inl
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set (Set Ω)
S T : Set ι
h_indep : iIndepSets s κ μ
hST : Disjoint S T
t1 t2 : Set Ω
p1 : Finset ι
hp1 : ↑p1 ⊆ S
f1 : ι → Set Ω
ht1_m : ∀ x ∈ p1, f1 x ∈ s x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
p2 : Finset ι
hp2 : ↑p2 ⊆ T
f2 : ι → Set Ω
ht2_m : ∀ x ∈ p2, f2 x ∈ s x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
g : ι → Set Ω := fun i ↦ (if i ∈ p1 then f1 i else univ) ∩ if i ∈ p2 then f2 i else univ
i : ι
hi1 : i ∈ p1
hi2 : i ∉ p2
⊢ f1 i ∩ univ ∈ s i
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
this : IsProbabilityMeasure μ
val✝ : Fintype ι
g : ι → Set Ω := fun i' ↦ if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
⊢ (μ (⋂ i, Y i ⁻¹' t i))⁻¹ * μ (⋂ i, g i) = (∏ i, μ (Y i ⁻¹' t i))⁻¹ * μ (⋂ i, g i)

Tactic:
rw [hindep.meas_iInter]

Goal state after:
ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
this : IsProbabilityMeasure μ
val✝ : Fintype ι
g : ι → Set Ω := fun i' ↦ if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
⊢ ∀ (i : ι), MeasurableSet (Y i ⁻¹' t i)
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
this : IsProbabilityMeasure μ
val✝ : Fintype ι
g : ι → Set Ω := fun i' ↦ if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
⊢ (∏ i, μ (Y i ⁻¹' t i))⁻¹ * μ (⋂ i, g i) = (∏ i, μ (Y i ⁻¹' t i))⁻¹ * ∏ i, μ (g i)

Tactic:
rw [hindep.meas_iInter]

Goal state after:
ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
this : IsProbabilityMeasure μ
val✝ : Fintype ι
g : ι → Set Ω := fun i' ↦ if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
⊢ ∀ (i : ι), MeasurableSet (g i)
Module: Mathlib.Probability.Density

Goal state before:
pp: case refine_1
Ω : Type u_1
E : Type u_2
inst✝ : MeasurableSpace E
x✝ : MeasurableSpace Ω
X : Ω → E
ℙ : Measure Ω
μ : Measure E
hX : AEMeasurable X ℙ
f : E → ℝ≥0∞
hf : AEMeasurable f μ
h : map X ℙ = μ.withDensity f
⊢ (μ.withDensity f).HaveLebesgueDecomposition μ

pp: case refine_2
Ω : Type u_1
E : Type u_2
inst✝ : MeasurableSpace E
x✝ : MeasurableSpace Ω
X : Ω → E
ℙ : Measure Ω
μ : Measure E
hX : AEMeasurable X ℙ
f : E → ℝ≥0∞
hf : AEMeasurable f μ
h : map X ℙ = μ.withDensity f
⊢ μ.withDensity f ≪ μ

Tactic:
· rw [withDensity_congr_ae hf.ae_eq_mk]
    exact haveLebesgueDecomposition_withDensity μ hf.measurable_mk

Goal state after:
case refine_2
Ω : Type u_1
E : Type u_2
inst✝ : MeasurableSpace E
x✝ : MeasurableSpace Ω
X : Ω → E
ℙ : Measure Ω
μ : Measure E
hX : AEMeasurable X ℙ
f : E → ℝ≥0∞
hf : AEMeasurable f μ
h : map X ℙ = μ.withDensity f
⊢ μ.withDensity f ≪ μ