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

Goal state before:
pp: case pos.h
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
hiff : M ^ (1 / p.toReal) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ p.toReal‖₊
hx : x ∈ {x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}
⊢ M ^ (1 / p.toReal) ≤ ↑‖f x‖₊

Tactic:
]

Goal state after:
case pos.h
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
hiff : M ^ (1 / p.toReal) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ p.toReal‖₊
hx : x ∈ {x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}
⊢ M ^ (1 / p.toReal) ≤ ↑‖f x‖₊
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
i : Set α
hi : MeasurableSet i
⊢ μ i + ν i = μ i + ν i

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
i : Set α
hi : MeasurableSet i
⊢ μ.real i - ν.real i = μ.real i - ν.real i

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
v : VectorMeasure α ℝ≥0∞
s : Set α
hs : MeasurableSet s
⊢ v s = v s

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : VectorMeasure α ℝ≥0∞
s : Set α
hs : MeasurableSet s
⊢ μ s = μ s

Tactic:
]

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

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

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
⊢ { re := 0, im := 0 } = 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
⊢ { re := 0, im := 0 } = 0
Module: Mathlib.MeasureTheory.Measure.Complex

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
i : Set α
hi : ¬MeasurableSet i
⊢ { re := 0, im := 0 } = 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
i : Set α
hi : ¬MeasurableSet i
⊢ { re := 0, im := 0 } = 0
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut

Goal state before:
pp: Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
inst✝¹ : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝ : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg : Integrable (fun ω ↦ (B (f ω)) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : ⋃ i, sets i = Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), μ (sets x) < ∞
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ ↑x
this : ∀ᵐ (a : Ω) ∂μ, ∀ (i : ℕ), a ∈ sets i → μ[fun ω ↦ (B (f ω)) (g ω) | m] a = (B (f a)) (μ[g | m] a)
⊢ μ[fun ω ↦ (B (f ω)) (g ω) | m] =ᵐ[μ] fun ω ↦ (B (f ω)) (μ[g | m] ω)

Tactic:
]

Goal state after:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
inst✝¹ : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝ : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg : Integrable (fun ω ↦ (B (f ω)) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : ⋃ i, sets i = Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), μ (sets x) < ∞
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ ↑x
this : ∀ᵐ (a : Ω) ∂μ, ∀ (i : ℕ), a ∈ sets i → μ[fun ω ↦ (B (f ω)) (g ω) | m] a = (B (f a)) (μ[g | m] a)
⊢ μ[fun ω ↦ (B (f ω)) (g ω) | m] =ᵐ[μ] fun ω ↦ (B (f ω)) (μ[g | m] ω)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case e'_5
α : Type u_1
β : Type u_2
m mα : MeasurableSpace α
inst✝² : MeasurableSpace β
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v✝ v : VectorMeasure α M
f : α → β
hf : Measurable f
g : ℕ → Set β
hg₁ : ∀ (i : ℕ), MeasurableSet (g i)
hg₂ : Pairwise (Disjoint on g)
x✝ : ℕ
⊢ v (f ⁻¹' g x✝) = v (f ⁻¹' g x✝)

Tactic:
]

Goal state after:
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
hiff : M ^ (1 / p.toReal) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ p.toReal‖₊
hx : x ∉ {x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}
⊢ 0 ^ p.toReal = 0

pp: case neg.h
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
hiff : M ^ (1 / p.toReal) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ p.toReal‖₊
hx : x ∉ {x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}
⊢ x ∉ {x | M ≤ ↑‖‖f x‖ ^ 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
hiff : M ^ (1 / p.toReal) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ p.toReal‖₊
hx : x ∉ {x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}
⊢ 0 ^ p.toReal = 0

case neg.h
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
hiff : M ^ (1 / p.toReal) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ p.toReal‖₊
hx : x ∉ {x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}
⊢ x ∉ {x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut

Goal state before:
pp: Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
inst✝¹ : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝ : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg : Integrable (fun ω ↦ (B (f ω)) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : ⋃ i, sets i = Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), μ (sets x) < ∞
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ ↑x
this : ∀ᵐ (a : Ω) ∂μ, ∀ (i : ℕ), a ∈ sets i → μ[fun ω ↦ (B (f ω)) (g ω) | m] a = (B (f a)) (μ[g | m] a)
ω : Ω
hω : ∀ (i : ℕ), ω ∈ sets i → μ[fun ω ↦ (B (f ω)) (g ω) | m] ω = (B (f ω)) (μ[g | m] ω)
⊢ ω ∈ Set.univ

Tactic:
]

Goal state after:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
inst✝¹ : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝ : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg : Integrable (fun ω ↦ (B (f ω)) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : ⋃ i, sets i = Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), μ (sets x) < ∞
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ ↑x
this : ∀ᵐ (a : Ω) ∂μ, ∀ (i : ℕ), a ∈ sets i → μ[fun ω ↦ (B (f ω)) (g ω) | m] a = (B (f a)) (μ[g | m] a)
ω : Ω
hω : ∀ (i : ℕ), ω ∈ sets i → μ[fun ω ↦ (B (f ω)) (g ω) | m] ω = (B (f ω)) (μ[g | m] ω)
⊢ ω ∈ Set.univ
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case e'_6
α : Type u_1
β : Type u_2
m mα : MeasurableSpace α
inst✝² : MeasurableSpace β
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v✝ v : VectorMeasure α M
f : α → β
hf : Measurable f
g : ℕ → Set β
hg₁ : ∀ (i : ℕ), MeasurableSet (g i)
hg₂ : Pairwise (Disjoint on g)
⊢ v (⋃ i, f ⁻¹' g i) = v (⋃ i, f ⁻¹' g i)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝² : MeasurableSpace β
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ { measureOf' := fun s ↦ if MeasurableSet s then v (f ⁻¹' s) else 0, empty' := ⋯, not_measurable' := ⋯,
        m_iUnion' := ⋯ }
      s =
    v (f ⁻¹' s)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝² : MeasurableSpace β
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ { measureOf' := fun s ↦ if MeasurableSet s then v (f ⁻¹' s) else 0, empty' := ⋯, not_measurable' := ⋯,
        m_iUnion' := ⋯ }
      s =
    v (f ⁻¹' s)
Module: Mathlib.MeasureTheory.Measure.Complex

Goal state before:
pp: case mpr
α : Type u_1
m : MeasurableSpace α
c : ComplexMeasure α
μ : VectorMeasure α ℝ≥0∞
h : re c ≪ᵥ μ ∧ im c ≪ᵥ μ
i : Set α
hi : μ i = 0
⊢ ↑0 + ↑0 * Complex.I = 0

pp: α : Type u_1
m : MeasurableSpace α
c : ComplexMeasure α
μ : VectorMeasure α ℝ≥0∞
h : re c ≪ᵥ μ ∧ im c ≪ᵥ μ
i : Set α
hi : μ i = 0
⊢ (c i).im = 0

pp: α : Type u_1
m : MeasurableSpace α
c : ComplexMeasure α
μ : VectorMeasure α ℝ≥0∞
h : re c ≪ᵥ μ ∧ im c ≪ᵥ μ
i : Set α
hi : μ i = 0
⊢ (c i).re = 0

Tactic:
]

Goal state after:
case mpr
α : Type u_1
m : MeasurableSpace α
c : ComplexMeasure α
μ : VectorMeasure α ℝ≥0∞
h : re c ≪ᵥ μ ∧ im c ≪ᵥ μ
i : Set α
hi : μ i = 0
⊢ ↑0 + ↑0 * Complex.I = 0

α : Type u_1
m : MeasurableSpace α
c : ComplexMeasure α
μ : VectorMeasure α ℝ≥0∞
h : re c ≪ᵥ μ ∧ im c ≪ᵥ μ
i : Set α
hi : μ i = 0
⊢ (c i).im = 0

α : Type u_1
m : MeasurableSpace α
c : ComplexMeasure α
μ : VectorMeasure α ℝ≥0∞
h : re c ≪ᵥ μ ∧ im c ≪ᵥ μ
i : Set α
hi : μ i = 0
⊢ (c i).re = 0
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
i : Set α
hi : MeasurableSet i
⊢ v i = v i

Tactic:
]

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

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝² : MeasurableSpace β
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
f : α → β
hf : Measurable f
i : Set β
hi : MeasurableSet i
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case neg.h
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
hiff : M ^ (1 / p.toReal) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ p.toReal‖₊
hx : x ∉ {x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}
⊢ ¬M ≤ ↑‖‖f x‖ ^ p.toReal‖₊

Tactic:
]

Goal state after:
case neg.h
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
hiff : M ^ (1 / p.toReal) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ p.toReal‖₊
hx : x ∉ {x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}
⊢ ¬M ≤ ↑‖‖f x‖ ^ p.toReal‖₊
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
m mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
v✝ : VectorMeasure α M
N : Type u_4
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
f : M →+ N
hf : Continuous[inst✝², inst✝] ⇑f
⊢ f (v ∅) = 0

Tactic:
empty,

Goal state after:
α : Type u_1
β : Type u_2
m mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
v✝ : VectorMeasure α M
N : Type u_4
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
f : M →+ N
hf : Continuous[inst✝², inst✝] ⇑f
⊢ f 0 = 0
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
m mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
v✝ : VectorMeasure α M
N : Type u_4
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
f : M →+ N
hf : Continuous[inst✝², inst✝] ⇑f
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case neg.h
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
hiff : M ^ (1 / p.toReal) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ p.toReal‖₊
hx : x ∉ {x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}
⊢ ¬M ^ (1 / p.toReal) ≤ ↑‖f x‖₊

Tactic:
]

Goal state after:
case neg.h
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
hiff : M ^ (1 / p.toReal) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ p.toReal‖₊
hx : x ∉ {x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}
⊢ ¬M ^ (1 / p.toReal) ≤ ↑‖f x‖₊
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
⊢ s.ExistsOneDivLT i (Nat.find ⋯)

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ¬s ≤[i] 0
⊢ s.ExistsOneDivLT i (Nat.find ⋯)
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
m : ℕ
hm : m < Nat.find ⋯
⊢ ¬s.ExistsOneDivLT i m

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ¬s ≤[i] 0
m : ℕ
hm : m < Nat.find ⋯
⊢ ¬s.ExistsOneDivLT i m
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
⊢ Classical.choose ⋯ ⊆ i ∧
    MeasurableSet (Classical.choose ⋯) ∧ 1 / (↑(s.findExistsOneDivLT i) + 1) < s (Classical.choose ⋯)

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ¬s ≤[i] 0
⊢ Classical.choose ⋯ ⊆ i ∧
    MeasurableSet (Classical.choose ⋯) ∧ 1 / (↑(s.findExistsOneDivLT i) + 1) < s (Classical.choose ⋯)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
m mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
v✝ : VectorMeasure α M
N : Type u_4
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
f : M →+ N
hf : Continuous[inst✝², inst✝] ⇑f
i : Set α
hi : ¬MeasurableSet i
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ¬¬s ≤[i] 0
⊢ ∅ ⊆ i

Tactic:
]

Goal state after:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ¬¬s ≤[i] 0
⊢ ∅ ⊆ i
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ¬¬s ≤[i] 0
⊢ MeasurableSet ∅

Tactic:
]

Goal state after:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ¬¬s ≤[i] 0
⊢ MeasurableSet ∅
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 : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hM : eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε
x : α
⊢ {x | M ≤ ↑‖f x‖₊}.indicator (fun a ↦ ‖f a‖) x ≤ {x | M ≤ ↑‖f x‖₊}.indicator (fun a ↦ ‖f a‖) x

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n : ℕ
⊢ s.someExistsOneDivLT
      (i \
        ⋃ k,
          ⋃ (H : k ≤ n),
            have this := ⋯;
            s.restrictNonposSeq i k) =
    s.someExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)

Tactic:
]

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

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ∞
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : M ≤ 0
s : Set α
x✝¹ : MeasurableSet s
x✝ : μ s ≤ ENNReal.ofReal 1
⊢ eLpNorm (s.indicator 0) p μ ≤ ENNReal.ofReal ε

pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ∞
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : M ≤ 0
s : Set α
x✝¹ : MeasurableSet s
x✝ : μ s ≤ ENNReal.ofReal 1
⊢ f = 0

Tactic:
]

Goal state after:
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ∞
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : M ≤ 0
s : Set α
x✝¹ : MeasurableSet s
x✝ : μ s ≤ ENNReal.ofReal 1
⊢ eLpNorm (s.indicator 0) p μ ≤ ENNReal.ofReal ε

α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ∞
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : M ≤ 0
s : Set α
x✝¹ : MeasurableSet s
x✝ : μ s ≤ ENNReal.ofReal 1
⊢ f = 0
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: case zero
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
⊢ s.someExistsOneDivLT (i \ ∅) ⊆ i

Tactic:
]

Goal state after:
case zero
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
⊢ s.someExistsOneDivLT (i \ ∅) ⊆ i
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: case succ
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n✝ : ℕ
⊢ s.someExistsOneDivLT
      (i \
        ⋃ k,
          ⋃ (H : k ≤ n✝),
            have this := ⋯;
            s.restrictNonposSeq i k) ⊆
    i

Tactic:
]

Goal state after:
case succ
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n✝ : ℕ
⊢ s.someExistsOneDivLT
      (i \
        ⋃ k,
          ⋃ (H : k ≤ n✝),
            have this := ⋯;
            s.restrictNonposSeq i k) ⊆
    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
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ∞
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : M ≤ 0
s : Set α
x✝¹ : MeasurableSet s
x✝ : μ s ≤ ENNReal.ofReal 1
x : α
⊢ ‖f x‖ ≤ 0

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ∞
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : M ≤ 0
s : Set α
x✝¹ : MeasurableSet s
x✝ : μ s ≤ ENNReal.ofReal 1
x : α
⊢ ‖f x‖ ≤ 0
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n : ℕ
hn : ¬s ≤[i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k] 0
⊢ 1 / (↑(s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)) + 1) <
    s (s.someExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k))

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n : ℕ
hn : ¬s ≤[i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k] 0
⊢ 1 / (↑(s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)) + 1) <
    s (s.someExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k))
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case e'_5
α : Type u_1
β : Type u_2
m mα : MeasurableSpace α
inst✝² : MeasurableSpace β
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v✝ v : VectorMeasure α M
i : Set α
hi : MeasurableSet i
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
x✝ : ℕ
⊢ v (f x✝ ∩ i) = v (f x✝ ∩ i)

Tactic:
]

Goal state after:
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
hp_top : p ≠ ∞
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
s : Set α
hs : MeasurableSet s
hμ : μ s ≤ ENNReal.ofReal ((ε / M) ^ p.toReal)
hp : ¬p = 0
⊢ eLpNorm f p (μ.restrict s) ≤ ENNReal.ofReal ε

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
hp_top : p ≠ ∞
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
s : Set α
hs : MeasurableSet s
hμ : μ s ≤ ENNReal.ofReal ((ε / M) ^ p.toReal)
hp : ¬p = 0
⊢ eLpNorm f p (μ.restrict s) ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: case zero
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₂ : ¬s ≤[i] 0
hn : ¬s ≤[i \ ⋃ k, ⋃ (_ : k < 0), s.restrictNonposSeq i k] 0
⊢ 0 < s (s.someExistsOneDivLT (i \ ∅))

Tactic:
]

Goal state after:
case zero
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₂ : ¬s ≤[i] 0
hn : ¬s ≤[i \ ⋃ k, ⋃ (_ : k < 0), s.restrictNonposSeq i k] 0
⊢ 0 < s (s.someExistsOneDivLT (i \ ∅))
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case e'_6
α : Type u_1
β : Type u_2
m mα : MeasurableSpace α
inst✝² : MeasurableSpace β
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v✝ v : VectorMeasure α M
i : Set α
hi : MeasurableSet i
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
⊢ v (⋃ i_1, f i_1 ∩ i) = v (⋃ i_1, f i_1 ∩ i)

Tactic:
]

Goal state after: