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

Goal state before:
pp: case intro.refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : AddCommMonoid M
inst✝³ : PartialOrder M
inst✝² : IsOrderedAddMonoid M
inst✝¹ : OrderClosedTopology M
v w : VectorMeasure α M
inst✝ : Countable β
f : β → Set α
hf₁ : ∀ (b : β), MeasurableSet (f b)
hf₂ : ∀ (b : β), v ≤[f b] w
val✝ : Encodable β
⊢ ∀ (n : ℕ), MeasurableSet (⋃ b ∈ Encodable.decode₂ β n, f b)

Tactic:
intro n

Goal state after:
case intro.refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : AddCommMonoid M
inst✝³ : PartialOrder M
inst✝² : IsOrderedAddMonoid M
inst✝¹ : OrderClosedTopology M
v w : VectorMeasure α M
inst✝ : Countable β
f : β → Set α
hf₁ : ∀ (b : β), MeasurableSet (f b)
hf₂ : ∀ (b : β), v ≤[f b] w
val✝ : Encodable β
n : ℕ
⊢ MeasurableSet (⋃ b ∈ Encodable.decode₂ β n, f b)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case intro.refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : AddCommMonoid M
inst✝³ : PartialOrder M
inst✝² : IsOrderedAddMonoid M
inst✝¹ : OrderClosedTopology M
v w : VectorMeasure α M
inst✝ : Countable β
f : β → Set α
hf₁ : ∀ (b : β), MeasurableSet (f b)
hf₂ : ∀ (b : β), v ≤[f b] w
val✝ : Encodable β
⊢ ∀ (n : ℕ), v ≤[⋃ b ∈ Encodable.decode₂ β n, f b] w

Tactic:
intro n

Goal state after:
case intro.refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : AddCommMonoid M
inst✝³ : PartialOrder M
inst✝² : IsOrderedAddMonoid M
inst✝¹ : OrderClosedTopology M
v w : VectorMeasure α M
inst✝ : Countable β
f : β → Set α
hf₁ : ∀ (b : β), MeasurableSet (f b)
hf₂ : ∀ (b : β), v ≤[f b] w
val✝ : Encodable β
n : ℕ
⊢ v ≤[⋃ b ∈ Encodable.decode₂ β n, f b] w
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
A : Set α := i \ ⋃ l, s.restrictNonposSeq i l
hA : A = i \ ⋃ l, s.restrictNonposSeq i l
bdd : ℕ → ℕ := fun n ↦ s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)
⊢ ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), s.restrictNonposSeq i l] 0

Tactic:
intro n

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
A : Set α := i \ ⋃ l, s.restrictNonposSeq i l
hA : A = i \ ⋃ l, s.restrictNonposSeq i l
bdd : ℕ → ℕ := fun n ↦ s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)
n : ℕ
⊢ ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), s.restrictNonposSeq i l] 0
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : AddCommMonoid N
inst✝¹ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
inst✝ : MeasureSpace β
h : v ≪ᵥ w
f : α → β
hf : ¬Measurable f
⊢ v.map f ≪ᵥ w.map f

Tactic:
intro s _

Goal state after:
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : AddCommMonoid N
inst✝¹ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
inst✝ : MeasureSpace β
h : v ≪ᵥ w
f : α → β
hf : ¬Measurable f
s : Set β
a✝ : (w.map f) s = 0
⊢ (v.map f) s = 0
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → w t = 0
t : Set α
hst : t ⊆ s
ht : MeasurableSet t
⊢ v t = 0

Tactic:
exact h₁ t hst ht

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ r • s = - -(r • s)

Tactic:
neg_neg

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ r • s = r • s
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
h : ∀ (x : ℝ), ∃ x_1 ∈ s.measureOfNegatives, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ s.measureOfNegatives ∧ f n < -↑n
⊢ ∀ (n : ℕ), ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -↑n

Tactic:
intro n

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
h : ∀ (x : ℝ), ∃ x_1 ∈ s.measureOfNegatives, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ s.measureOfNegatives ∧ f n < -↑n
n : ℕ
⊢ ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -↑n
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
h : ∀ (x : ℝ), ∃ x_1 ∈ s.measureOfNegatives, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ s.measureOfNegatives ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), s ≤[B n] 0
h_lt : ∀ (n : ℕ), s (B n) < -↑n
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
⊢ ∀ (n : ℕ), s A ≤ -↑n

Tactic:
intro n

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
h : ∀ (x : ℝ), ∃ x_1 ∈ s.measureOfNegatives, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ s.measureOfNegatives ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), s ≤[B n] 0
h_lt : ∀ (n : ℕ), s (B n) < -↑n
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
n : ℕ
⊢ s A ≤ -↑n
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
h : ∀ (x : ℝ), ∃ x_1 ∈ s.measureOfNegatives, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ s.measureOfNegatives ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), s ≤[B n] 0
h_lt : ∀ (n : ℕ), s (B n) < -↑n
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
n : ℕ
⊢ s ((⋃ i, B i) \ B n) + s (B n) ≤ s (B n)

pp: α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
h : ∀ (x : ℝ), ∃ x_1 ∈ s.measureOfNegatives, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ s.measureOfNegatives ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), s ≤[B n] 0
h_lt : ∀ (n : ℕ), s (B n) < -↑n
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
n : ℕ
⊢ MeasurableSet ((⋃ i, B i) \ B n)

Tactic:
· refine add_le_of_nonpos_left ?_
      have : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hmeas hr
      refine nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ ?_ Set.sdiff_subset this)
      exact MeasurableSet.iUnion hmeas

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
h : ∀ (x : ℝ), ∃ x_1 ∈ s.measureOfNegatives, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ s.measureOfNegatives ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), s ≤[B n] 0
h_lt : ∀ (n : ℕ), s (B n) < -↑n
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
n : ℕ
⊢ MeasurableSet ((⋃ i, B i) \ B n)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : 0 ≤[i] s
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
h₁ : ∀ (n : ℕ), MeasurableSet (i ∩ f n)
⊢ Pairwise (Disjoint on fun n ↦ i ∩ f n)

Tactic:
intro n

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : 0 ≤[i] s
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
h₁ : ∀ (n : ℕ), MeasurableSet (i ∩ f n)
n : ℕ
⊢ ∀ ⦃j : ℕ⦄, n ≠ j → (Disjoint on fun n ↦ i ∩ f n) n j
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
s : SignedMeasure α
μ : Measure α
⊢ (s.singularPart μ).totalVariation =
    s.toJordanDecomposition.posPart.singularPart μ + s.toJordanDecomposition.negPart.singularPart μ

Tactic:
by
  have :
    (s.singularPart μ).toJordanDecomposition =
      ⟨s.toJordanDecomposition.posPart.singularPart μ,
        s.toJordanDecomposition.negPart.singularPart μ, singularPart_mutuallySingular s μ⟩ := by
    refine JordanDecomposition.toSignedMeasure_injective ?_
    rw [toSignedMeasure_toJordanDecomposition, singularPart, JordanDecomposition.toSignedMeasure]
  rw [totalVariation, this]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
ι : Type u_3
s : Finset ι
f : ι → α → E
hf : ∀ i ∈ s, MemLp (f i) p μ
hp : 1 ≤ p
⊢ lpNorm (∑ i ∈ s, f i) p μ ≤ ∑ i ∈ s, (eLpNorm (f i) p μ).toReal

Tactic:
← ENNReal.toReal_sum fun i hi ↦ (hf i hi).2.ne,

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
ι : Type u_3
s : Finset ι
f : ι → α → E
hf : ∀ i ∈ s, MemLp (f i) p μ
hp : 1 ≤ p
⊢ lpNorm (∑ i ∈ s, f i) p μ ≤ (∑ a ∈ s, eLpNorm (f a) p μ).toReal
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : Module ℚ≥0 E
inst✝ : NormedSpace ℝ E
ι : Type u_3
s : Finset ι
f : ι → α → E
hf : ∀ i ∈ s, MemLp (f i) p μ
hp : 1 ≤ p
⊢ lpNorm (𝔼 i ∈ s, f i) p μ ≤ 𝔼 i ∈ s, lpNorm (f i) p μ

Tactic:
obtain rfl | hs := s.eq_empty_or_nonempty
  · simp
  refine (le_inv_smul_iff_of_pos <| by positivity).2 ?_
  rw [Nat.cast_smul_eq_nsmul, ← lpNorm_nsmul, Finset.card_smul_expect]
  exact lpNorm_sum_le hf hp

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
K : Type u_3
inst✝ : RCLike K
f : α → K
p : ℝ≥0∞
μ : Measure α
hf : ¬AEStronglyMeasurable f μ
h : AEStronglyMeasurable ((starRingEnd (α → K)) f) μ
⊢ AEStronglyMeasurable f μ

Tactic:
simpa [Function.comp_def]
      using (continuous_star.measurable.comp_aemeasurable h.aemeasurable).aestronglyMeasurable

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

Goal state before:
pp: case inr
α : Type u_1
m : MeasurableSpace α
s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
⊢ - -r •
      ((s.toJordanDecomposition.posPart.singularPart μ).toSignedMeasure -
        (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure) =
    r •
      ((s.toJordanDecomposition.posPart.singularPart μ).toSignedMeasure -
        (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure)

Tactic:
neg_neg

Goal state after:
case inr
α : Type u_1
m : MeasurableSpace α
s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
⊢ r •
      ((s.toJordanDecomposition.posPart.singularPart μ).toSignedMeasure -
        (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure) =
    r •
      ((s.toJordanDecomposition.posPart.singularPart μ).toSignedMeasure -
        (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure)
Module: Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Goal state before:
pp: Ω : Type u_1
inst✝ : MeasurableSpace Ω
μ : ProbabilityMeasure Ω
f : ℕ → Set Ω
hf : Summable fun n ↦ μ (f n)
⊢ μ (⋃ n, f n) ≤ ∑' (n : ℕ), μ (f n)

Tactic:
by
  simpa [← ENNReal.coe_le_coe, ENNReal.coe_tsum hf] using MeasureTheory.measure_iUnion_le f

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∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i ↦ Exists.choose ⋯
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hCg : ∀ (i : ι), eLpNorm ({x | C ≤ ‖g i x‖₊}.indicator (g i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
⊢ ‖{x | max C 1 ≤ ‖Exists.choose ⋯ x‖₊}.indicator (Exists.choose ⋯) x‖ ≤ ‖{x | C ≤ ‖g i x‖₊}.indicator (g i) x‖

Tactic:
norm_indicator_eq_indicator_norm,

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∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i ↦ Exists.choose ⋯
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hCg : ∀ (i : ι), eLpNorm ({x | C ≤ ‖g i x‖₊}.indicator (g i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
⊢ {x | max C 1 ≤ ‖Exists.choose ⋯ x‖₊}.indicator (fun a ↦ ‖Exists.choose ⋯ a‖) x ≤
    ‖{x | C ≤ ‖g i x‖₊}.indicator (g i) x‖
Module: Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Goal state before:
pp: Ω : Type u_1
inst✝ : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
nonzero : μ ≠ 0
⊢ μ.normalize.toFiniteMeasure = μ.mass⁻¹ • μ

Tactic:
μ.self_eq_mass_smul_normalize

Goal state after:
Ω : Type u_1
inst✝ : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
nonzero : μ ≠ 0
⊢ μ.normalize.toFiniteMeasure = μ.mass⁻¹ • μ.mass • μ.normalize.toFiniteMeasure
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: Ω : Type u_1
inst✝ : MeasurableSpace Ω
μ : FiniteMeasure Ω
f : ℕ → Set Ω
hf : Summable fun n ↦ μ (f n)
⊢ μ (⋃ n, f n) ≤ ∑' (n : ℕ), μ (f n)

Tactic:
by
  simpa [← ENNReal.coe_le_coe, ENNReal.coe_tsum hf] using MeasureTheory.measure_iUnion_le f

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

Goal state before:
pp: Ω : Type u_1
inst✝ : Nonempty Ω
m0 : MeasurableSpace Ω
μ : ProbabilityMeasure Ω
s : Set Ω
_s_mble : MeasurableSet s
⊢ μ.toFiniteMeasure.mass⁻¹ * μ.toFiniteMeasure s = μ s

Tactic:
simp only [ProbabilityMeasure.mass_toFiniteMeasure, inv_one, one_mul, μ.coeFn_toFiniteMeasure]

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

Goal state before:
pp: Ω : Type u_1
inst✝¹ : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝ : TopologicalSpace Ω
f : Ω →ᵇ ℝ≥0
⊢ μ.testAgainstNN f = μ.mass * μ.normalize.toFiniteMeasure.testAgainstNN f

Tactic:
μ.self_eq_mass_smul_normalize

Goal state after:
Ω : Type u_1
inst✝¹ : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝ : TopologicalSpace Ω
f : Ω →ᵇ ℝ≥0
⊢ (μ.mass • μ.normalize.toFiniteMeasure).testAgainstNN f = μ.mass * μ.normalize.toFiniteMeasure.testAgainstNN f
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : BorelSpace Ω
μ ν : FiniteMeasure Ω
hμν : μ.toWeakDualBCNN = ν.toWeakDualBCNN
f : Ω →ᵇ ℝ≥0
⊢ ∫⁻ (x : Ω), ↑(f x) ∂↑μ = ∫⁻ (x : Ω), ↑(f x) ∂↑ν

Tactic:
have key := congr_fun (congrArg DFunLike.coe hμν) f

Goal state after:
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : BorelSpace Ω
μ ν : FiniteMeasure Ω
hμν : μ.toWeakDualBCNN = ν.toWeakDualBCNN
f : Ω →ᵇ ℝ≥0
key : μ.toWeakDualBCNN f = ν.toWeakDualBCNN f
⊢ ∫⁻ (x : Ω), ↑(f x) ∂↑μ = ∫⁻ (x : Ω), ↑(f x) ∂↑ν
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : IsFiniteMeasure μ
g : α → ℝ
hint : Integrable g μ
ℱ : ι → MeasurableSpace α
hℱ : ∀ (i : ι), ℱ i ≤ m0
A : MeasurableSpace α := m0
hmeas : ∀ (n : ι) (C : ℝ≥0), MeasurableSet {x | C ≤ ‖μ[g | ℱ n] x‖₊}
hg : MemLp g 1 μ
ε : ℝ
hε : 0 < ε
hne : ¬eLpNorm g 1 μ = 0
δ : ℝ
hδ : 0 < δ
h : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator g) 1 μ ≤ ENNReal.ofReal ε
C : ℝ≥0 := (NNReal.mk δ ⋯)⁻¹ * (eLpNorm g 1 μ).toNNReal
hC : C = (NNReal.mk δ ⋯)⁻¹ * (eLpNorm g 1 μ).toNNReal
hCpos : 0 < C
⊢ ∀ (n : ι), μ {x | C ≤ ‖μ[g | ℱ n] x‖₊} ≤ ENNReal.ofReal δ

Tactic:
intro n

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : IsFiniteMeasure μ
g : α → ℝ
hint : Integrable g μ
ℱ : ι → MeasurableSpace α
hℱ : ∀ (i : ι), ℱ i ≤ m0
A : MeasurableSpace α := m0
hmeas : ∀ (n : ι) (C : ℝ≥0), MeasurableSet {x | C ≤ ‖μ[g | ℱ n] x‖₊}
hg : MemLp g 1 μ
ε : ℝ
hε : 0 < ε
hne : ¬eLpNorm g 1 μ = 0
δ : ℝ
hδ : 0 < δ
h : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator g) 1 μ ≤ ENNReal.ofReal ε
C : ℝ≥0 := (NNReal.mk δ ⋯)⁻¹ * (eLpNorm g 1 μ).toNNReal
hC : C = (NNReal.mk δ ⋯)⁻¹ * (eLpNorm g 1 μ).toNNReal
hCpos : 0 < C
n : ι
⊢ μ {x | C ≤ ‖μ[g | ℱ n] x‖₊} ≤ ENNReal.ofReal δ
Module: Mathlib.MeasureTheory.Measure.Tight

Goal state before:
pp: case inl
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
r : ENNReal := μ univ
hεr : ε < r
hεr' : r - ε < r
K : Set 𝓧
left✝ : K ⊆ univ
hKμ : r - ε < μ K
hK_compact : IsCompact K
hK_closed : IsClosed[inst✝²] K
⊢ μ Kᶜ ≤ ε

Tactic:
measure_compl hK_closed.measurableSet (measure_ne_top μ _),

Goal state after:
case inl
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
r : ENNReal := μ univ
hεr : ε < r
hεr' : r - ε < r
K : Set 𝓧
left✝ : K ⊆ univ
hKμ : r - ε < μ K
hK_compact : IsCompact K
hK_closed : IsClosed[inst✝²] K
⊢ μ univ - μ K ≤ ε
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
s' : Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
u : Set ι
hyp : ∀ n ∈ u, IndepSets (s n) s' κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ ⋃ n ∈ u, s n
ht2 : t2 ∈ s'
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2

Tactic:
simp_rw [Set.mem_iUnion] at ht1

Goal state after:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
s' : Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
u : Set ι
hyp : ∀ n ∈ u, IndepSets (s n) s' κ μ
t1 t2 : Set Ω
ht2 : t2 ∈ s'
ht1 : ∃ i, ∃ (_ : i ∈ u), t1 ∈ s i
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2
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
s : ι → Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h_indep : iIndepSets s κ μ
i j : ι
hij : i ≠ j
t₁ t₂ : Set Ω
ht₁ : t₁ ∈ s i
ht₂ : t₂ ∈ s j
x : ι
hx✝ : x ∈ {i, j}
hx : x = i
⊢ (if x = i then t₁ else t₂) ∈ s x

pp: case inr
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h_indep : iIndepSets s κ μ
i j : ι
hij : i ≠ j
t₁ t₂ : Set Ω
ht₁ : t₁ ∈ s i
ht₂ : t₂ ∈ s j
x : ι
hx✝ : x ∈ {i, j}
hx : x ∈ {j}
⊢ (if x = i then t₁ else t₂) ∈ s x

Tactic:
· simp [hx, ht₁]

Goal state after:
case inr
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h_indep : iIndepSets s κ μ
i j : ι
hij : i ≠ j
t₁ t₂ : Set Ω
ht₁ : t₁ ∈ s i
ht₂ : t₂ ∈ s j
x : ι
hx✝ : x ∈ {i, j}
hx : x ∈ {j}
⊢ (if x = i then t₁ else t₂) ∈ s x
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
m : (i : ι) → MeasurableSpace (β i)
f✝ : (i : ι) → Ω → β i
h : ∀ (s : Finset ι), iIndepFun (s.restrict f✝) μ
s : Finset ι
f : ι → Set Ω
hs : ∀ i ∈ s, MeasurableSet (f i)
this : ⋂ i ∈ s, f i = ⋂ i, f ↑i
⊢ μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i)

Tactic:
← Finset.prod_coe_sort,

Goal state after:
Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
β : ι → Type u_10
m : (i : ι) → MeasurableSpace (β i)
f✝ : (i : ι) → Ω → β i
h : ∀ (s : Finset ι), iIndepFun (s.restrict f✝) μ
s : Finset ι
f : ι → Set Ω
hs : ∀ i ∈ s, MeasurableSet (f i)
this : ⋂ i ∈ s, f i = ⋂ i, f ↑i
⊢ μ (⋂ i ∈ s, f i) = ∏ i, μ (f ↑i)
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
h : Measure.map (fun ω i ↦ f i ω) μ = Measure.pi fun i ↦ Measure.map (f i) μ
S : Finset ι
s : (i : ι) → Set (β i)
hs : ∀ i ∈ S, MeasurableSet (s i)
h₀ :
  ∏ x ∈ S, μ (f x ⁻¹' s x) = (∏ x ∈ S, (Measure.map (f x) μ) (s x)) * ∏ x with x ∉ S, (Measure.map (f x) μ) univ ∧
    μ (⋂ i ∈ S, f i ⁻¹' s i) = (∏ x ∈ S, (Measure.map (f x) μ) (s x)) * ∏ x with x ∉ S, (Measure.map (f x) μ) univ
⊢ (∏ x ∈ S, (Measure.map (f x) μ) (s x)) * ∏ x with x ∉ S, (Measure.map (f x) μ) univ = ∏ i ∈ S, μ (f i ⁻¹' s i)

Tactic:
← h₀.1

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
h : Measure.map (fun ω i ↦ f i ω) μ = Measure.pi fun i ↦ Measure.map (f i) μ
S : Finset ι
s : (i : ι) → Set (β i)
hs : ∀ i ∈ S, MeasurableSet (s i)
h₀ :
  ∏ x ∈ S, μ (f x ⁻¹' s x) = (∏ x ∈ S, (Measure.map (f x) μ) (s x)) * ∏ x with x ∉ S, (Measure.map (f x) μ) univ ∧
    μ (⋂ i ∈ S, f i ⁻¹' s i) = (∏ x ∈ S, (Measure.map (f x) μ) (s x)) * ∏ x with x ∉ S, (Measure.map (f x) μ) univ
⊢ ∏ x ∈ S, μ (f x ⁻¹' s x) = ∏ i ∈ S, μ (f i ⁻¹' s i)
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: 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
h_P_inter✝ : ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = ∏ n ∈ p1 ∪ p2, (κ a) (g n)
a : α
h_P_inter : (κ a) (t1 ∩ t2) = ∏ n ∈ p1 ∪ p2, (κ a) (g n)
ha1 : (κ a) (⋂ i ∈ p1, f1 i) = ∏ i ∈ p1, (κ a) (f1 i)
ha2 : (κ a) (⋂ i ∈ p2, f2 i) = ∏ i ∈ p2, (κ a) (f2 i)
h' : IsProbabilityMeasure (κ a)
⊢ ∀ (n : ι), (κ a) (g n) = (if n ∈ p1 then (κ a) (f1 n) else 1) * if n ∈ p2 then (κ a) (f2 n) else 1

Tactic:
intro n

Goal state after:
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
h_P_inter✝ : ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = ∏ n ∈ p1 ∪ p2, (κ a) (g n)
a : α
h_P_inter : (κ a) (t1 ∩ t2) = ∏ n ∈ p1 ∪ p2, (κ a) (g n)
ha1 : (κ a) (⋂ i ∈ p1, f1 i) = ∏ i ∈ p1, (κ a) (f1 i)
ha2 : (κ a) (⋂ i ∈ p2, f2 i) = ∏ i ∈ p2, (κ a) (f2 i)
h' : IsProbabilityMeasure (κ a)
n : ι
⊢ (κ a) (g n) = (if n ∈ p1 then (κ a) (f1 n) else 1) * if n ∈ p2 then (κ a) (f2 n) else 1