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

Goal state before:
pp: case refine_2
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝¹ : CompleteSpace E
hm : m ≤ mΩ
inst✝ : IsFiniteMeasure μ
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (ω : Ω) ∂μ, ‖f ω‖ ≤ c
fs : ℕ → SimpleFunc Ω F := hf.approxBounded c
hfs_tendsto : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun x ↦ (fs x) ω) atTop (𝓝 (f ω))
hμ : ¬μ = 0
this : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (ω : Ω), ‖(fs n) ω‖ ≤ c
⊢ Integrable (fun ω ↦ (B (f ω)) (μ[g | m] ω)) μ

Tactic:
·

Goal state after:
case refine_2
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝¹ : CompleteSpace E
hm : m ≤ mΩ
inst✝ : IsFiniteMeasure μ
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (ω : Ω) ∂μ, ‖f ω‖ ≤ c
fs : ℕ → SimpleFunc Ω F := hf.approxBounded c
hfs_tendsto : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun x ↦ (fs x) ω) atTop (𝓝 (f ω))
hμ : ¬μ = 0
this : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (ω : Ω), ‖(fs n) ω‖ ≤ c
⊢ Integrable (fun ω ↦ (B (f ω)) (μ[g | m] ω)) μ
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
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
n : ℕ
x : α
hx : ↑n ≤ ‖f x‖
⊢ x ∈ {x | (fun a ↦ ‖{x | ↑n ≤ ‖f x‖₊}.indicator f a‖ ≤ ‖f a‖) x}

pp: case neg
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
n : ℕ
x : α
hx : ¬↑n ≤ ‖f x‖
⊢ x ∈ {x | (fun a ↦ ‖{x | ↑n ≤ ‖f x‖₊}.indicator f a‖ ≤ ‖f a‖) x}

Tactic:
·

Goal state after:
case pos
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
n : ℕ
x : α
hx : ↑n ≤ ‖f x‖
⊢ x ∈ {x | (fun a ↦ ‖{x | ↑n ≤ ‖f x‖₊}.indicator f a‖ ≤ ‖f a‖) x}
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
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
n : ℕ
x : α
hx : ¬↑n ≤ ‖f x‖
⊢ x ∈ {x | (fun a ↦ ‖{x | ↑n ≤ ‖f x‖₊}.indicator f a‖ ≤ ‖f a‖) x}

Tactic:
·

Goal state after:
case neg
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
n : ℕ
x : α
hx : ¬↑n ≤ ‖f x‖
⊢ x ∈ {x | (fun a ↦ ‖{x | ↑n ≤ ‖f x‖₊}.indicator f a‖ ≤ ‖f a‖) x}
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
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
n : ℕ
x : α
hx : ¬↑n ≤ ‖f x‖
⊢ 0 ≤ ‖f x‖

pp: case neg.h
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
n : ℕ
x : α
hx : ¬↑n ≤ ‖f x‖
⊢ x ∉ {x | ↑n ≤ ‖f x‖₊}

Tactic:
·

Goal state after:
case neg
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
n : ℕ
x : α
hx : ¬↑n ≤ ‖f x‖
⊢ 0 ≤ ‖f x‖
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
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
n : ℕ
x : α
hx : ¬↑n ≤ ‖f x‖
⊢ x ∉ {x | ↑n ≤ ‖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
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
n : ℕ
x : α
hx : ¬↑n ≤ ‖f x‖
⊢ x ∉ {x | ↑n ≤ ‖f x‖₊}
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: case inl
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
he : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1
⊢ e ∈ transvections K V ^ (finrank K (V ⧸ (↑e).fixedSubmodule) - 1) * dilatransvections K V

pp: case inr.inl
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
he : e.fixedReduce = 1
⊢ e ∈ transvections K V ^ (finrank K (V ⧸ (↑e).fixedSubmodule) - 1) * dilatransvections K V

pp: case inr.inr
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
he : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x
⊢ e ∈ transvections K V ^ (finrank K (V ⧸ (↑e).fixedSubmodule) - 1) * dilatransvections K V

Tactic:
·

Goal state after:
case inl
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
he : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1
⊢ e ∈ transvections K V ^ (finrank K (V ⧸ (↑e).fixedSubmodule) - 1) * dilatransvections K V
Module: Mathlib.MeasureTheory.Covering.LiminfLimsup

Goal state before:
pp: case inl
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
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
R₁ : ℕ → ℝ := fun i ↦ max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i ↦ max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
hM' : 1 ≤ M
⊢ blimsup (fun i ↦ cthickening (max 0 (r₁ i)) (s i)) atTop p ≤ᵐ[μ]
    blimsup (fun i ↦ cthickening (max 0 (r₂ i)) (s i)) atTop p

pp: case inr
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
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
R₁ : ℕ → ℝ := fun i ↦ max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i ↦ max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
hM' : M < 1
⊢ blimsup (fun i ↦ cthickening (max 0 (r₁ i)) (s i)) atTop p ≤ᵐ[μ]
    blimsup (fun i ↦ cthickening (max 0 (r₂ i)) (s i)) atTop p

Tactic:
·

Goal state after:
case inl
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
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
R₁ : ℕ → ℝ := fun i ↦ max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i ↦ max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
hM' : 1 ≤ M
⊢ blimsup (fun i ↦ cthickening (max 0 (r₁ i)) (s i)) atTop p ≤ᵐ[μ]
    blimsup (fun i ↦ cthickening (max 0 (r₂ i)) (s i)) atTop p
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: case inr.inl
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
he : e.fixedReduce = 1
⊢ e ∈ transvections K V ^ (finrank K (V ⧸ (↑e).fixedSubmodule) - 1) * dilatransvections K V

pp: case inr.inr
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
he : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x
⊢ e ∈ transvections K V ^ (finrank K (V ⧸ (↑e).fixedSubmodule) - 1) * dilatransvections K V

Tactic:
·

Goal state after:
case inr.inl
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
he : e.fixedReduce = 1
⊢ e ∈ transvections K V ^ (finrank K (V ⧸ (↑e).fixedSubmodule) - 1) * dilatransvections K V
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: case inr.inr
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
he : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x
⊢ e ∈ transvections K V ^ (finrank K (V ⧸ (↑e).fixedSubmodule) - 1) * dilatransvections K V

Tactic:
·

Goal state after:
case inr.inr
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
he : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x
⊢ e ∈ transvections K V ^ (finrank K (V ⧸ (↑e).fixedSubmodule) - 1) * dilatransvections K V
Module: Mathlib.MeasureTheory.Covering.LiminfLimsup

Goal state before:
pp: case inr
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
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
R₁ : ℕ → ℝ := fun i ↦ max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i ↦ max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
hM' : M < 1
⊢ blimsup (fun i ↦ cthickening (max 0 (r₁ i)) (s i)) atTop p ≤ᵐ[μ]
    blimsup (fun i ↦ cthickening (max 0 (r₂ i)) (s i)) atTop p

Tactic:
·

Goal state after:
case inr
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
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
R₁ : ℕ → ℝ := fun i ↦ max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i ↦ max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
hM' : M < 1
⊢ blimsup (fun i ↦ cthickening (max 0 (r₁ i)) (s i)) atTop p ≤ᵐ[μ]
    blimsup (fun i ↦ cthickening (max 0 (r₂ i)) (s i)) atTop p
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

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

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) → v = w

Tactic:
·

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

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) → v = w
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

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

pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v w : VectorMeasure α M
h : ∀ (i : Set α), MeasurableSet i → v i = w i
i : Set α
hi : ¬MeasurableSet i
⊢ v i = w i

Tactic:
·

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

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

Tactic:
·

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

Goal state before:
pp: case e'_5
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
x✝ : ℕ
⊢ Function.extend e (fun i ↦ v (f i)) 0 x✝ = v (Function.extend e f (fun x ↦ ∅) x✝)

pp: case e'_6
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
⊢ ⋃ i, f i = ⋃ i, Function.extend e f (fun x ↦ ∅) i

pp: case convert_1
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
⊢ ∀ (i : ℕ), MeasurableSet (Function.extend e f (fun x ↦ ∅) i)

pp: case convert_2
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
⊢ Pairwise (Disjoint on Function.extend e f fun x ↦ ∅)

Tactic:
·

Goal state after:
case e'_5
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
x✝ : ℕ
⊢ Function.extend e (fun i ↦ v (f i)) 0 x✝ = v (Function.extend e f (fun x ↦ ∅) x✝)
Module: Mathlib.MeasureTheory.Covering.LiminfLimsup

Goal state before:
pp: case refine_1
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)
⊢ blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i ↦ cthickening (r i) (s i)) atTop p

pp: case refine_2
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)
⊢ blimsup (fun i ↦ cthickening (r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop p

Tactic:
·

Goal state after:
case refine_1
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)
⊢ blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i ↦ cthickening (r i) (s i)) atTop p
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case e'_6
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
⊢ ⋃ i, f i = ⋃ i, Function.extend e f (fun x ↦ ∅) i

pp: case convert_1
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
⊢ ∀ (i : ℕ), MeasurableSet (Function.extend e f (fun x ↦ ∅) i)

pp: case convert_2
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
⊢ Pairwise (Disjoint on Function.extend e f fun x ↦ ∅)

Tactic:
·

Goal state after:
case e'_6
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
⊢ ⋃ i, f i = ⋃ i, Function.extend e f (fun x ↦ ∅) 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
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case convert_1
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
⊢ ∀ (i : ℕ), MeasurableSet (Function.extend e f (fun x ↦ ∅) i)

pp: case convert_2
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
⊢ Pairwise (Disjoint on Function.extend e f fun x ↦ ∅)

Tactic:
·

Goal state after:
case convert_1
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
⊢ ∀ (i : ℕ), MeasurableSet (Function.extend e f (fun x ↦ ∅) i)
Module: Mathlib.MeasureTheory.Covering.LiminfLimsup

Goal state before:
pp: case refine_2
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)
⊢ blimsup (fun i ↦ cthickening (r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop p

Tactic:
·

Goal state after:
case refine_2
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)
⊢ blimsup (fun i ↦ cthickening (r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop p
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case convert_2
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
⊢ Pairwise (Disjoint on Function.extend e f fun x ↦ ∅)

Tactic:
·

Goal state after:
case convert_2
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
⊢ Pairwise (Disjoint on Function.extend e f fun x ↦ ∅)