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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ∞
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hi : eLpNorm (fun x ↦ f x - y₀) p μ < ∞
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x ↦ ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal
h_bound : ∀ (n : ℕ), (fun x ↦ ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal) ≤ᵐ[μ] fun x ↦ ‖f x - y₀‖ₑ ^ p.toReal
h_fin : ∫⁻ (a : β), ‖f a - y₀‖ₑ ^ p.toReal ∂μ ≠ ∞
⊢ ∀ᵐ (a : β) ∂μ, Tendsto (fun x ↦ ‖(approxOn f hf s y₀ h₀ x) a - f a‖ₑ ^ p.toReal) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
T : Set α → β
inst✝ : AddMonoid β
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : FinMeasAdditive (c • μ) T
⊢ FinMeasAdditive μ T

Tactic:
by

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

Goal state before:
pp: X : Type u_1
E : Type u_6
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : NormedAddCommGroup E
μ : Measure X
inst✝⁶ : OpensMeasurableSpace X
inst✝⁵ : LocallyCompactSpace X
inst✝⁴ : T2Space X
𝕜 : Type u_9
inst✝³ : NormedRing 𝕜
inst✝² : SecondCountableTopologyEither X 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : X → E
g : X → 𝕜
s : Set X
hs : IsLocallyClosed s
hf : LocallyIntegrableOn f s μ
hg : ContinuousOn g s
⊢ LocallyIntegrableOn (fun x ↦ g x • f x) s μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
T : Set α → β
inst✝ : AddMonoid β
c : ℝ≥0∞
hc_ne_zero : c ≠ 0
hT : FinMeasAdditive μ T
⊢ FinMeasAdditive (c • μ) T

Tactic:
by

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ∞
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hi : eLpNorm (fun x ↦ f x - y₀) p μ < ∞
⊢ Tendsto (fun n ↦ eLpNorm (⇑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_8
inst✝ : AddCancelMonoid β
T : Set α → β
hT : FinMeasAdditive μ T
⊢ T ∅ = 0

Tactic:
by

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

Goal state before:
pp: X : Type u_1
E : Type u_6
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : NormedAddCommGroup E
μ : Measure X
inst✝⁶ : OpensMeasurableSpace X
inst✝⁵ : LocallyCompactSpace X
inst✝⁴ : T2Space X
𝕜 : Type u_9
inst✝³ : NormedRing 𝕜
inst✝² : SecondCountableTopologyEither X E
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : X → 𝕜
g : X → E
s : Set X
hs : IsLocallyClosed s
hf : LocallyIntegrableOn f s μ
hg : ContinuousOn g s
⊢ LocallyIntegrableOn (fun x ↦ f x • g x) s μ

Tactic:
by

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f p μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x ↦ y₀) p μ
n : ℕ
this : eLpNorm (fun x ↦ (approxOn f fmeas s y₀ h₀ n) x - y₀) p μ < ∞
⊢ eLpNorm (⇑(approxOn f fmeas s y₀ h₀ n)) p μ < ∞

Tactic:
by

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f p μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x ↦ y₀) p μ
n : ℕ
⊢ Measurable fun x ↦ ‖f x - y₀‖

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
T T' : Set α → β
inst✝ : AddCommMonoid β
hT : FinMeasAdditive μ T
hT' : FinMeasAdditive μ T'
⊢ FinMeasAdditive μ (T + T')

Tactic:
by

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f p μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x ↦ y₀) p μ
n : ℕ
⊢ MemLp (fun x ↦ ‖f x - y₀‖) p μ

Tactic:
by

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f p μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x ↦ y₀) p μ
n : ℕ
hf' : MemLp (fun x ↦ ‖f x - y₀‖) p μ
⊢ ∀ᵐ (x : β) ∂μ, ‖(approxOn f fmeas s y₀ h₀ n) x - y₀‖ ≤ ‖‖f x - y₀‖ + ‖f x - y₀‖‖

Tactic:
by

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f p μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x ↦ y₀) p μ
n : ℕ
⊢ MemLp (⇑(approxOn f fmeas s y₀ h₀ n)) p μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
q : ℝ
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
hp_ne_top : p ≠ ∞
μ : Measure β
fmeas : Measurable f
inst✝ : SeparableSpace ↑(Set.range f ∪ {0})
hf : eLpNorm f p μ < ∞
n : ℕ
⊢ 0 ∈ Set.range f ∪ {0}

Tactic:
by

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
hp_ne_top : p ≠ ∞
μ : Measure β
fmeas : Measurable f
inst✝ : SeparableSpace ↑(Set.range f ∪ {0})
hf : eLpNorm f p μ < ∞
x : β
⊢ f x ∈ Set.range f ∪ {0}

Tactic:
by

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
hp_ne_top : p ≠ ∞
μ : Measure β
fmeas : Measurable f
inst✝ : SeparableSpace ↑(Set.range f ∪ {0})
hf : eLpNorm f p μ < ∞
⊢ Tendsto (fun n ↦ eLpNorm (⇑(approxOn f fmeas (Set.range f ∪ {0}) 0 ⋯ n) - f) p μ) atTop (𝓝 0)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
q : ℝ
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
inst✝ : SeparableSpace ↑(Set.range f ∪ {0})
hf : MemLp f p μ
n : ℕ
⊢ 0 ∈ Set.range f ∪ {0}

Tactic:
by

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
inst✝ : SeparableSpace ↑(Set.range f ∪ {0})
hf : MemLp f p μ
n : ℕ
⊢ 0 ∈ Set.range f ∪ {0}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
q : ℝ
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
μ : Measure β
fmeas : Measurable f
inst✝ : SeparableSpace ↑(Set.range f ∪ {0})
hf : MemLp f p μ
n : ℕ
⊢ 0 ∈ Set.range f ∪ {0}

Tactic:
by

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
μ : Measure β
fmeas : Measurable f
inst✝ : SeparableSpace ↑(Set.range f ∪ {0})
hf : MemLp f p μ
⊢ Tendsto (fun n ↦ MemLp.toLp ⇑(approxOn f fmeas (Set.range f ∪ {0}) 0 ⋯ n) ⋯) atTop (𝓝 (MemLp.toLp f hf))

Tactic:
by

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

Goal state before:
pp: β : Type u_2
inst✝¹ : MeasurableSpace β
p : ℝ≥0∞
E : Type u_7
inst✝ : NormedAddCommGroup E
f : β → E
μ : Measure β
hf : MemLp f p μ
hp_ne_top : p ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
f' : β → E := AEStronglyMeasurable.mk f ⋯
g : β →ₛ E
hg : eLpNorm (f' - ⇑g) p μ < ε
g_mem : MemLp (⇑g) p μ
this : eLpNorm (f - ⇑g) p μ = eLpNorm (f' - ⇑g) p μ
⊢ eLpNorm (f - ⇑g) p μ < ε

Tactic:
by

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

Goal state before:
pp: β : Type u_2
inst✝¹ : MeasurableSpace β
p : ℝ≥0∞
E : Type u_7
inst✝ : NormedAddCommGroup E
f : β → E
μ : Measure β
hf : MemLp f p μ
hp_ne_top : p ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ∃ g, eLpNorm (f - ⇑g) p μ < ε ∧ MemLp (⇑g) p μ

Tactic:
by

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hi : HasFiniteIntegral (fun x ↦ f x - y₀) μ
⊢ eLpNorm (fun x ↦ f x - y₀) 1 μ < ∞

Tactic:
by

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hi : HasFiniteIntegral (fun x ↦ f x - y₀) μ
⊢ Tendsto (fun n ↦ ∫⁻ (x : β), ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ∂μ) atTop (𝓝 0)

Tactic:
by

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : Integrable f μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : Integrable (fun x ↦ y₀) μ
n : ℕ
⊢ Integrable (⇑(approxOn f fmeas s y₀ h₀ n)) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : OpensMeasurableSpace E
f : β → E
μ : Measure β
inst✝ : SeparableSpace ↑(Set.range f ∪ {0})
fmeas : Measurable f
hf : Integrable f μ
n : ℕ
x : β
⊢ 0 ∈ Set.range f ∪ {0}

Tactic:
by

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : OpensMeasurableSpace E
f : β → E
μ : Measure β
inst✝ : SeparableSpace ↑(Set.range f ∪ {0})
fmeas : Measurable f
hf : Integrable f μ
x : β
⊢ f x ∈ Set.range f ∪ {0}

Tactic:
by

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : OpensMeasurableSpace E
f : β → E
μ : Measure β
inst✝ : SeparableSpace ↑(Set.range f ∪ {0})
fmeas : Measurable f
hf : Integrable f μ
⊢ Tendsto (fun n ↦ ∫⁻ (x : β), ‖(approxOn f fmeas (Set.range f ∪ {0}) 0 ⋯ n) x - f x‖ₑ ∂μ) atTop (𝓝 0)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
inst✝ : SeparableSpace ↑(Set.range f ∪ {0})
hf : Integrable f μ
n : ℕ
⊢ 0 ∈ Set.range f ∪ {0}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
F : Type u_5
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup F
p : ℝ
f : α →ₛ F
μ : Measure α
⊢ (fun x ↦ ‖f x‖ₑ ^ p) = ⇑(map (fun x ↦ ‖x‖ₑ ^ p) f)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
F : Type u_5
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup F
p : ℝ
f : α →ₛ F
μ : Measure α
⊢ eLpNorm' (⇑f) p μ = (∑ y ∈ f.range, ‖y‖ₑ ^ p * μ (⇑f ⁻¹' {y})) ^ (1 / p)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
f : α →ₛ E
hf : MemLp (⇑f) p μ
y : E
hy_ne : y ≠ 0
⊢ (map (fun x ↦ ‖x‖ₑ ^ p.toReal) f).FinMeasSupp μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
f : α →ₛ E
hf : MemLp (⇑f) p μ
y : E
hy_ne : y ≠ 0
h_fin : (map (fun x ↦ ‖x‖ₑ ^ p.toReal) f).FinMeasSupp μ
b : E
⊢ (fun x ↦ ‖x‖ₑ ^ p.toReal) b = 0 ↔ b = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
f : α →ₛ E
hf : MemLp (⇑f) p μ
y : E
hy_ne : y ≠ 0
h_fin : (map (fun x ↦ ‖x‖ₑ ^ p.toReal) f).FinMeasSupp μ
⊢ f.FinMeasSupp μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
f : α →ₛ E
hf : MemLp (⇑f) p μ
y : E
hy_ne : y ≠ 0
⊢ μ (⇑f ⁻¹' {y}) < ∞

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
Fₗ : Type u_6
inst✝¹⁷ : AddCommGroup E
inst✝¹⁶ : UniformSpace E
inst✝¹⁵ : IsUniformAddGroup E
inst✝¹⁴ : AddCommGroup F
inst✝¹³ : UniformSpace F
inst✝¹² : IsUniformAddGroup F
inst✝¹¹ : T0Space F
inst✝¹⁰ : AddCommMonoid Eₗ
inst✝⁹ : UniformSpace Eₗ
inst✝⁸ : ContinuousAdd Eₗ
inst✝⁷ : Semiring 𝕜
inst✝⁶ : Semiring 𝕜₂
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : Module 𝕜 Eₗ
inst✝² : ContinuousConstSMul 𝕜 Eₗ
inst✝¹ : ContinuousConstSMul 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
f g : E →SL[σ₁₂] F
inst✝ : CompleteSpace F
e : E →L[𝕜] Eₗ
h : DenseRange ⇑e ∧ IsUniformInducing ⇑e
cont : Continuous (⋯.extend ⇑f)
eq : ∀ (b : E), ⋯.extend (⇑f) (e b) = f b
⊢ ∀ (x y : Eₗ), ⋯.extend (⇑f) (x + y) = ⋯.extend (⇑f) x + ⋯.extend (⇑f) y

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ 0 ≤ 1 / p.toReal

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
⊢ MemLp (⇑f) p μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_4
F : Type u_5
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
μ : Measure α
f : α →ₛ E
g : α →ₛ F
⊢ Integrable (⇑f) μ → Integrable (⇑g) μ → Integrable (⇑(f.pair g)) μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
hSp : ∀ i ∈ sι, μ (S i) ≠ ∞
h_disj : ∀ i ∈ sι, ∀ j ∈ sι, i ≠ j → Disjoint (S i) (S j)
⊢ T (⋃ i ∈ sι, S i) = ∑ i ∈ sι, T (S i)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
c : E
hc : c ≠ 0
s : Set α
hs : MeasurableSet s
hcs : MemLp (⇑(piecewise s hs (const α c) (const α 0))) p μ
⊢ μ s < ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
⊢ ∀ {a b : ↥(Lp E p μ)},
    a ∈ {f | ∃ s, AEEqFun.mk ⇑s ⋯ = ↑f} → b ∈ {f | ∃ s, AEEqFun.mk ⇑s ⋯ = ↑f} → a + b ∈ {f | ∃ s, AEEqFun.mk ⇑s ⋯ = ↑f}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
⊢ ∀ {x : ↥(Lp E p μ)}, x ∈ {f | ∃ s, AEEqFun.mk ⇑s ⋯ = ↑f} → -x ∈ {f | ∃ s, AEEqFun.mk ⇑s ⋯ = ↑f}

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
T : Set α → β
inst✝ : AddGroup β
hT : FinMeasAdditive μ T
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
hμs : μ s ≠ ∞
hμt : μ t ≠ ∞
hst : Disjoint s t
⊢ T s + T t = T t + T s

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
T : Set α → β
inst✝ : AddGroup β
hT : FinMeasAdditive μ T
⊢ FinMeasAdditive μ (-T)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
k : 𝕜
f : ↥(simpleFunc E p μ)
⊢ k • ↑f ∈ simpleFunc E p μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
x y : 𝕜
f : ↥(simpleFunc E p μ)
⊢ (x * y) • f = x • y • f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
β : Type u_7
inst✝ : SeminormedAddCommGroup β
C : ℝ
m : MeasurableSpace α
μ : Measure α
hC : 0 ≤ C
⊢ DominatedFinMeasAdditive μ 0 C

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : ↥(simpleFunc E p μ)
⊢ 1 • f = f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_8
inst✝ : NormedAddCommGroup β
T : Set α → β
C : ℝ
hT : DominatedFinMeasAdditive μ T C
s : Set α
hs : MeasurableSet s
hs_zero : μ s = 0
⊢ μ s < ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
x : 𝕜
⊢ x • 0 = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
x : 𝕜
f g : ↥(simpleFunc E p μ)
⊢ x • (f + g) = x • f + x • g

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
x y : 𝕜
f : ↥(simpleFunc E p μ)
⊢ (x + y) • f = x • f + y • f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_8
inst✝ : NormedAddCommGroup β
T : Set α → β
C : ℝ
hT : DominatedFinMeasAdditive μ T C
s : Set α
hs : MeasurableSet s
hs_zero : μ s = 0
⊢ T s = 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
β : Type u_8
inst✝ : NormedAddCommGroup β
T : Set α → β
C : ℝ
x✝ : MeasurableSpace α
hT : DominatedFinMeasAdditive 0 T C
s : Set α
hs : MeasurableSet s
⊢ 0 s = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : ↥(simpleFunc E p μ)
⊢ 0 • f = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : α →ₛ E
hf : MemLp (⇑f) p μ
hg : MemLp (⇑g) p μ
⊢ (f - g).toLp ⋯ = f.toLp hf - g.toLp hg

Tactic:
by

Goal state after: