mathlib-initiative/mathlib-tactics huggingface dataset:
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
f : α → β
hf : MemLp f 1 μ
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M ↦ {x | ↑M ≤ ↑‖f x‖₊}.indicator f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable ({x | ↑M ≤ ↑‖f x‖₊}.indicator f) μ
hbound : HasFiniteIntegral (fun x ↦ ‖f x‖) μ
this : ∀ ε > 0, ∃ N, ∀ n ≥ N, ∫⁻ (a : α), ENNReal.ofReal ‖{x | ↑n ≤ ‖f x‖₊}.indicator f a - 0‖ ∂μ ≤ ε
⊢ ∃ M, ∫⁻ (x : α), ↑‖{x | M ≤ ↑‖f x‖₊}.indicator f x‖₊ ∂μ ≤ ENNReal.ofReal ε

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f 1 μ
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M ↦ {x | ↑M ≤ ↑‖f x‖₊}.indicator f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable ({x | ↑M ≤ ↑‖f x‖₊}.indicator f) μ
hbound : HasFiniteIntegral (fun x ↦ ‖f x‖) μ
this : ∀ ε > 0, ∃ N, ∀ n ≥ N, ∫⁻ (a : α), ENNReal.ofReal ‖{x | ↑n ≤ ‖f x‖₊}.indicator f a - 0‖ ∂μ ≤ ε
⊢ ∃ M, ∫⁻ (x : α), ↑‖{x | M ≤ ↑‖f x‖₊}.indicator f x‖₊ ∂μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case mpr
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v w : VectorMeasure α M
⊢ (∀ (i : Set α), MeasurableSet i → v i = w i) → ∀ (x : Set α), v x = w x

Tactic:
]

Goal state after:
case mpr
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v w : VectorMeasure α M
⊢ (∀ (i : Set α), MeasurableSet i → v i = w i) → ∀ (x : Set α), v x = w x
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : Countable β
v : VectorMeasure α M
f : β → Set α
hm : ∀ (i : β), MeasurableSet (f i)
hd : Pairwise (Disjoint on f)
e : β → ℕ
he : Function.Injective e
⊢ HasSum (Function.extend e (fun i ↦ v (f i)) 0) (v (⋃ i, f i))

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : Countable β
v : VectorMeasure α M
f : β → Set α
hm : ∀ (i : β), MeasurableSet (f i)
hd : Pairwise (Disjoint on f)
e : β → ℕ
he : Function.Injective e
⊢ HasSum (Function.extend e (fun i ↦ v (f i)) 0) (v (⋃ i, f i))
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
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
⊢ eLpNormEssSup f (μ.restrict {x | (eLpNorm f ∞ μ + 1).toReal ≤ ↑‖f x‖₊}) = 0

pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
⊢ MeasurableSet {x | (eLpNorm f ∞ μ + 1).toReal ≤ ↑‖f x‖₊}

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
⊢ eLpNormEssSup f (μ.restrict {x | (eLpNorm f ∞ μ + 1).toReal ≤ ↑‖f x‖₊}) = 0

α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
⊢ MeasurableSet {x | (eLpNorm f ∞ μ + 1).toReal ≤ ↑‖f x‖₊}
Module: Mathlib.MeasureTheory.Covering.LiminfLimsup

Goal state before:
pp: α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[>] 0)
hr' : Tendsto (fun i ↦ M * r i) atTop (𝓝[>] 0)
i : ℕ
⊢ r i ≤ r i

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Covering.LiminfLimsup

Goal state before:
pp: α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[>] 0)
hr' : Tendsto (fun i ↦ M * r i) atTop (𝓝[>] 0)
i : ℕ
⊢ M⁻¹ * (M * r i) ≤ r i

Tactic:
rw [inv_mul_cancel_left₀ hM.ne' (r i)]

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
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
x : α
hx : x ∈ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖}
⊢ (eLpNormEssSup f μ).toReal < ‖f x‖

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
x : α
hx : x ∈ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖}
⊢ (eLpNormEssSup f μ).toReal < ‖f x‖
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
ι : Type u_4
s : Set ι
f : ι → Set α
hs : s.Countable
hd : s.Pairwise (Disjoint on f)
h : ∀ b ∈ s, MeasurableSet (f b)
this : Encodable ↑s
⊢ v (⋃ x, f ↑x) = ∑' (p : ↑s), v (f ↑p)

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
ι : Type u_4
s : Set ι
f : ι → Set α
hs : s.Countable
hd : s.Pairwise (Disjoint on f)
h : ∀ b ∈ s, MeasurableSet (f b)
this : Encodable ↑s
⊢ v (⋃ x, f ↑x) = ∑' (p : ↑s), v (f ↑p)
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
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
x : α
hx : x ∈ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖}
⊢ eLpNormEssSup f μ < eLpNormEssSup f μ + 1

pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
x : α
hx : x ∈ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖}
⊢ eLpNormEssSup f μ + 1 ≠ ∞

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
x : α
hx : x ∈ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖}
⊢ eLpNormEssSup f μ < eLpNormEssSup f μ + 1

α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
x : α
hx : x ∈ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖}
⊢ eLpNormEssSup f μ + 1 ≠ ∞
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
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
this : {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖} ⊆ {x | eLpNormEssSup f μ < ↑‖f x‖₊}
⊢ μ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖} ≤ 0

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
this : {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖} ⊆ {x | eLpNormEssSup f μ < ↑‖f x‖₊}
⊢ μ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖} ≤ 0
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
ι : Type u_4
s : Finset ι
f : ι → Set α
hd : (↑s).PairwiseDisjoint f
hm : ∀ b ∈ s, MeasurableSet (f b)
⊢ v (⋃ b ∈ s, f b) = ∑' (b : ↥s), v (f ↑b)

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
ι : Type u_4
s : Finset ι
f : ι → Set α
hd : (↑s).PairwiseDisjoint f
hm : ∀ b ∈ s, MeasurableSet (f b)
⊢ v (⋃ b ∈ s, f b) = ∑' (b : ↥s), v (f ↑b)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
ι : Type u_4
s : Finset ι
f : ι → Set α
hd : (↑s).PairwiseDisjoint f
hm : ∀ b ∈ s, MeasurableSet (f b)
⊢ v (⋃ b ∈ s, f b) = ∑ p ∈ s, v (f p)

Tactic:
rw [← Finset.sum_attach, Finset.attach_eq_univ, ← tsum_fintype (L := .unconditional s)]
  exact of_biUnion s.countable_toSet hd hm

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
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
this : μ.restrict {x | (eLpNorm f ∞ μ + 1).toReal ≤ ↑‖f x‖₊} = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
h : Disjoint A B
hA : MeasurableSet A
hB : MeasurableSet B
⊢ v A + v B = v A + v B

pp: case hm
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
h : Disjoint A B
hA : MeasurableSet A
hB : MeasurableSet B
⊢ ∀ (i : Bool), MeasurableSet (bif i then A else B)

pp: case hd
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
h : Disjoint A B
hA : MeasurableSet A
hB : MeasurableSet B
⊢ Pairwise (Disjoint on fun b ↦ bif b then A else B)

Tactic:
]

Goal state after:
case hm
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
h : Disjoint A B
hA : MeasurableSet A
hB : MeasurableSet B
⊢ ∀ (i : Bool), MeasurableSet (bif i then A else B)

case hd
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
h : Disjoint A B
hA : MeasurableSet A
hB : MeasurableSet B
⊢ Pairwise (Disjoint on fun b ↦ bif b then A else B)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h : A ⊆ B
⊢ v B = v B

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Covering.LiminfLimsup

Goal state before:
pp: α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[>] 0) →
      blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i ↦ cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i ↦ if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[>] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
i : ℕ
hi : r i ≤ 0
hi' : M * r i ≤ 0
⊢ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s i) =
    closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s i)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h : A ⊆ B
⊢ v (B \ A) = v (B \ A)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ v (A \ B) + v (A ∩ B) = v (A \ B) + v (A ∩ B)

pp: case h
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ Disjoint (A \ B) (A ∩ B)

pp: case hA
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A \ B)

pp: case hB
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A ∩ B)

Tactic:
]

Goal state after:
case h
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ Disjoint (A \ B) (A ∩ B)

case hA
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A \ B)

case hB
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A ∩ B)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case h
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ Disjoint (A ∩ B) (A \ B)

Tactic:
]

Goal state after:
case h
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ Disjoint (A ∩ B) (A \ B)
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ∞
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ‖{x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indicator (fun x ↦ ‖f x‖ ^ p.toReal) x‖ₑ ∂μ ≤
    ENNReal.ofReal (ε ^ p.toReal)
⊢ (∫⁻ (x : α), ‖{x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}.indicator f x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal) ≤
    ENNReal.ofReal ε ^ 1

Tactic:
]

Goal state after:
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ∞
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ‖{x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indicator (fun x ↦ ‖f x‖ ^ p.toReal) x‖ₑ ∂μ ≤
    ENNReal.ofReal (ε ^ p.toReal)
⊢ (∫⁻ (x : α), ‖{x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}.indicator f x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal) ≤
    ENNReal.ofReal ε ^ 1
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ v (A \ B) + v (A ∩ B) = v (A \ B) + v (A ∩ B)

pp: case h
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ Disjoint (A ∩ B) (B \ A)

pp: case hA
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A ∩ B)

pp: case hB
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (B \ A)

Tactic:
]

Goal state after:
case h
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ Disjoint (A ∩ B) (B \ A)

case hA
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A ∩ B)

case hB
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (B \ A)
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ∞
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ‖{x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indicator (fun x ↦ ‖f x‖ ^ p.toReal) x‖ₑ ∂μ ≤
    ENNReal.ofReal (ε ^ p.toReal)
⊢ (∫⁻ (x : α), ‖{x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}.indicator f x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal) ≤
    (ENNReal.ofReal ε ^ p.toReal) ^ (1 / p.toReal)

Tactic:
]

Goal state after:
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ∞
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ‖{x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indicator (fun x ↦ ‖f x‖ ^ p.toReal) x‖ₑ ∂μ ≤
    ENNReal.ofReal (ε ^ p.toReal)
⊢ (∫⁻ (x : α), ‖{x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}.indicator f x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal) ≤
    (ENNReal.ofReal ε ^ p.toReal) ^ (1 / p.toReal)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ v (A \ B) + v B = v (A \ B) + v B

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Covering.LiminfLimsup

Goal state before:
pp: α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[>] 0) →
      blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i ↦ cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i ↦ if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[>] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i ↦ p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ ((blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop fun i ↦ p i ∧ 0 < r i) ⊔
      blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop fun i ↦ p i ∧ r i ≤ 0) =ᵐ[μ]
    (blimsup (fun i ↦ cthickening (r i) (s i)) atTop fun i ↦ p i ∧ 0 < r i) ⊔
      blimsup (fun i ↦ cthickening (r i) (s i)) atTop fun i ↦ p i ∧ r i ≤ 0

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[>] 0) →
      blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i ↦ cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i ↦ if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[>] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i ↦ p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ ((blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop fun i ↦ p i ∧ 0 < r i) ⊔
      blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop fun i ↦ p i ∧ r i ≤ 0) =ᵐ[μ]
    (blimsup (fun i ↦ cthickening (r i) (s i)) atTop fun i ↦ p i ∧ 0 < r i) ⊔
      blimsup (fun i ↦ cthickening (r i) (s i)) atTop fun i ↦ p i ∧ r i ≤ 0
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
s : SignedMeasure α
A B : Set α
h : Disjoint A B
hA₁ : MeasurableSet A
hB₁ : MeasurableSet B
hA₂ : 0 ≤ s A
hB₂ : 0 ≤ s B
hAB : s A + s B = 0
⊢ s A = 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
s : SignedMeasure α
A B : Set α
h : Disjoint A B
hA₁ : MeasurableSet A
hB₁ : MeasurableSet B
hA₂ : 0 ≤ s A
hB₂ : 0 ≤ s B
hAB : s A + s B = 0
⊢ s A = 0
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case h₁
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ∞
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ‖{x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indicator (fun x ↦ ‖f x‖ ^ p.toReal) x‖ₑ ∂μ ≤
    ENNReal.ofReal (ε ^ p.toReal)
⊢ ∫⁻ (x : α), ‖{x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}.indicator f x‖ₑ ^ p.toReal ∂μ ≤ ENNReal.ofReal (ε ^ p.toReal)

Tactic:
]

Goal state after:
case h₁
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ∞
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ‖{x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indicator (fun x ↦ ‖f x‖ ^ p.toReal) x‖ₑ ∂μ ≤
    ENNReal.ofReal (ε ^ p.toReal)
⊢ ∫⁻ (x : α), ‖{x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}.indicator f x‖ₑ ^ p.toReal ∂μ ≤ ENNReal.ofReal (ε ^ p.toReal)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
s : SignedMeasure α
A B : Set α
h : Disjoint A B
hA₁ : MeasurableSet A
hB₁ : MeasurableSet B
hA₂ : s A ≤ 0
hB₂ : s B ≤ 0
hAB : s A + s B = 0
⊢ s A = 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
s : SignedMeasure α
A B : Set α
h : Disjoint A B
hA₁ : MeasurableSet A
hB₁ : MeasurableSet B
hA₂ : s A ≤ 0
hB₂ : s B ≤ 0
hAB : s A + s B = 0
⊢ s A = 0