mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
f : α → ε'
hf : IntegrableOn f s μ
h's : ∀ x ∈ s, ‖f x‖ₑ ≠ 0
⊢ μ.restrict (toMeasurable μ s) = μ.restrict s

Tactic:
rcases exists_seq_strictAnti_tendsto' ENNReal.zero_lt_top with ⟨u, _, u_pos, u_lim⟩

Goal state after:
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
f : α → ε'
hf : IntegrableOn f s μ
h's : ∀ x ∈ s, ‖f x‖ₑ ≠ 0
u : ℕ → ℝ≥0∞
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), u n ∈ Ioo 0 ∞
u_lim : Tendsto u atTop (𝓝 0)
⊢ μ.restrict (toMeasurable μ s) = μ.restrict s
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
s t : Set α
μ : Measure α
ε' : Type u_7
inst✝² : TopologicalSpace ε'
inst✝¹ : ENormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
hf : IntegrableOn f s μ
ht : NullMeasurableSet t μ
h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0
u : Set α := {x | x ∈ s ∧ f x ≠ 0}
hu : IntegrableOn f u μ
v : Set α := toMeasurable μ u
⊢ Integrable f (μ.restrict u)

pp: α : Type u_1
mα : MeasurableSpace α
s t : Set α
μ : Measure α
ε' : Type u_7
inst✝² : TopologicalSpace ε'
inst✝¹ : ENormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
hf : IntegrableOn f s μ
ht : NullMeasurableSet t μ
h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0
u : Set α := {x | x ∈ s ∧ f x ≠ 0}
hu : IntegrableOn f u μ
v : Set α := toMeasurable μ u
⊢ ∀ x ∈ u, ‖f x‖ₑ ≠ 0

Tactic:
·

Goal state after:
α : Type u_1
mα : MeasurableSpace α
s t : Set α
μ : Measure α
ε' : Type u_7
inst✝² : TopologicalSpace ε'
inst✝¹ : ENormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
hf : IntegrableOn f s μ
ht : NullMeasurableSet t μ
h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0
u : Set α := {x | x ∈ s ∧ f x ≠ 0}
hu : IntegrableOn f u μ
v : Set α := toMeasurable μ u
⊢ Integrable f (μ.restrict u)
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
s t : Set α
μ : Measure α
ε' : Type u_7
inst✝² : TopologicalSpace ε'
inst✝¹ : ENormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
hf : IntegrableOn f s μ
ht : NullMeasurableSet t μ
h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0
u : Set α := {x | x ∈ s ∧ f x ≠ 0}
hu : IntegrableOn f u μ
v : Set α := toMeasurable μ u
⊢ ∀ x ∈ u, ‖f x‖ₑ ≠ 0

Tactic:
·

Goal state after:
α : Type u_1
mα : MeasurableSpace α
s t : Set α
μ : Measure α
ε' : Type u_7
inst✝² : TopologicalSpace ε'
inst✝¹ : ENormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
hf : IntegrableOn f s μ
ht : NullMeasurableSet t μ
h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0
u : Set α := {x | x ∈ s ∧ f x ≠ 0}
hu : IntegrableOn f u μ
v : Set α := toMeasurable μ u
⊢ ∀ x ∈ u, ‖f x‖ₑ ≠ 0
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousENorm E
f : α → E
hf : Integrable f μ
ε : ℝ≥0∞
hε : 0 < ε
hε' : ε ≠ ∞
⊢ ε⁻¹ ^ ENNReal.toReal 1 < ∞

pp: case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousENorm E
f : α → E
hf : Integrable f μ
ε : ℝ≥0∞
hε : 0 < ε
hε' : ε ≠ ∞
⊢ eLpNorm f 1 μ ^ ENNReal.toReal 1 < ∞

Tactic:
·

Goal state after:
case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousENorm E
f : α → E
hf : Integrable f μ
ε : ℝ≥0∞
hε : 0 < ε
hε' : ε ≠ ∞
⊢ ε⁻¹ ^ ENNReal.toReal 1 < ∞
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousENorm E
f : α → E
hf : Integrable f μ
ε : ℝ≥0∞
hε : 0 < ε
hε' : ε ≠ ∞
⊢ eLpNorm f 1 μ ^ ENNReal.toReal 1 < ∞

Tactic:
·

Goal state after:
case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousENorm E
f : α → E
hf : Integrable f μ
ε : ℝ≥0∞
hε : 0 < ε
hε' : ε ≠ ∞
⊢ eLpNorm f 1 μ ^ ENNReal.toReal 1 < ∞
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
s t : Set α
μ : Measure α
ε' : Type u_7
inst✝² : TopologicalSpace ε'
inst✝¹ : ENormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
hf : IntegrableOn f s μ
ht : NullMeasurableSet t μ
h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0
u : Set α := {x | x ∈ s ∧ f x ≠ 0}
hu : IntegrableOn f u μ
v : Set α := toMeasurable μ u
A : IntegrableOn f v μ
x : α
hxt : x ∈ t \ s → f x = 0
hx : x ∈ t \ v
h'x : x ∈ s
⊢ 0 = f x

pp: case neg
α : Type u_1
mα : MeasurableSpace α
s t : Set α
μ : Measure α
ε' : Type u_7
inst✝² : TopologicalSpace ε'
inst✝¹ : ENormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
hf : IntegrableOn f s μ
ht : NullMeasurableSet t μ
h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0
u : Set α := {x | x ∈ s ∧ f x ≠ 0}
hu : IntegrableOn f u μ
v : Set α := toMeasurable μ u
A : IntegrableOn f v μ
x : α
hxt : x ∈ t \ s → f x = 0
hx : x ∈ t \ v
h'x : x ∉ s
⊢ 0 = f x

Tactic:
·

Goal state after:
case pos
α : Type u_1
mα : MeasurableSpace α
s t : Set α
μ : Measure α
ε' : Type u_7
inst✝² : TopologicalSpace ε'
inst✝¹ : ENormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
hf : IntegrableOn f s μ
ht : NullMeasurableSet t μ
h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0
u : Set α := {x | x ∈ s ∧ f x ≠ 0}
hu : IntegrableOn f u μ
v : Set α := toMeasurable μ u
A : IntegrableOn f v μ
x : α
hxt : x ∈ t \ s → f x = 0
hx : x ∈ t \ v
h'x : x ∈ s
⊢ 0 = f x
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousENorm E
f : α → E
hf : Integrable f μ
ε : ℝ≥0∞
hε : 0 < ε
hε' : ε = ∞
⊢ μ {x | ε < ‖f x‖ₑ} < ∞

pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousENorm E
f : α → E
hf : Integrable f μ
ε : ℝ≥0∞
hε : 0 < ε
hε' : ¬ε = ∞
⊢ μ {x | ε < ‖f x‖ₑ} < ∞

Tactic:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousENorm E
f : α → E
hf : Integrable f μ
ε : ℝ≥0∞
hε : 0 < ε
hε' : ε = ∞
⊢ μ {x | ε < ‖f x‖ₑ} < ∞
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
s t : Set α
μ : Measure α
ε' : Type u_7
inst✝² : TopologicalSpace ε'
inst✝¹ : ENormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
hf : IntegrableOn f s μ
ht : NullMeasurableSet t μ
h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0
u : Set α := {x | x ∈ s ∧ f x ≠ 0}
hu : IntegrableOn f u μ
v : Set α := toMeasurable μ u
A : IntegrableOn f v μ
x : α
hxt : x ∈ t \ s → f x = 0
hx : x ∈ t \ v
h'x : x ∉ s
⊢ 0 = f x

Tactic:
·

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
s t : Set α
μ : Measure α
ε' : Type u_7
inst✝² : TopologicalSpace ε'
inst✝¹ : ENormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
hf : IntegrableOn f s μ
ht : NullMeasurableSet t μ
h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0
u : Set α := {x | x ∈ s ∧ f x ≠ 0}
hu : IntegrableOn f u μ
v : Set α := toMeasurable μ u
A : IntegrableOn f v μ
x : α
hxt : x ∈ t \ s → f x = 0
hx : x ∈ t \ v
h'x : x ∉ s
⊢ 0 = f x
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
s t : Set α
μ : Measure α
ε' : Type u_7
inst✝² : TopologicalSpace ε'
inst✝¹ : ENormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
hf : IntegrableOn f s μ
ht : NullMeasurableSet t μ
h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0
u : Set α := {x | x ∈ s ∧ f x ≠ 0}
hu : IntegrableOn f u μ
v : Set α := toMeasurable μ u
A : IntegrableOn f v μ
x : α
hxt : x ∈ t \ s → f x = 0
hx : x ∈ t \ v
h'x : x ∉ s
⊢ 0 = f x

Tactic:
exact (hxt ⟨hx.1, h'x⟩).symm

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
hs : Summable fun x ↦ ‖f x‖
hs' : (Function.support f).Countable
this✝ : MeasurableSpace β := borel β
this : BorelSpace β
⊢ AEMeasurable f Measure.count

pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
hs : Summable fun x ↦ ‖f x‖
hs' : (Function.support f).Countable
this✝ : MeasurableSpace β := borel β
this : BorelSpace β
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂Measure.count, f x ∈ t

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
hs : Summable fun x ↦ ‖f x‖
hs' : (Function.support f).Countable
this✝ : MeasurableSpace β := borel β
this : BorelSpace β
⊢ AEMeasurable f Measure.count
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
hs : Summable fun x ↦ ‖f x‖
hs' : (Function.support f).Countable
this✝ : MeasurableSpace β := borel β
this : BorelSpace β
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂Measure.count, f x ∈ t

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
hs : Summable fun x ↦ ‖f x‖
hs' : (Function.support f).Countable
this✝ : MeasurableSpace β := borel β
this : BorelSpace β
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂Measure.count, f x ∈ t
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g : α → E
H : AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ
⊢ Integrable g (μ.withDensity fun x ↦ ↑(f x)) ↔ Integrable (fun x ↦ ↑(f x) • g x) μ

pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g : α → E
H : ¬AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ
⊢ Integrable g (μ.withDensity fun x ↦ ↑(f x)) ↔ Integrable (fun x ↦ ↑(f x) • g x) μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g : α → E
H : AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ
⊢ Integrable g (μ.withDensity fun x ↦ ↑(f x)) ↔ Integrable (fun x ↦ ↑(f x) • g x) μ
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g : α → E
H : AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ
⊢ ∫⁻ (a : α), ((fun x ↦ ↑(f x)) * fun a ↦ ‖g a‖ₑ) a ∂μ < ∞ ↔ ∫⁻ (a : α), ‖↑(f a) • g a‖ₑ ∂μ < ∞

pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g : α → E
H : AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ
⊢ AEMeasurable (fun a ↦ ‖g a‖ₑ) (μ.withDensity fun x ↦ ↑(f x))

Tactic:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g : α → E
H : AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ
⊢ ∫⁻ (a : α), ((fun x ↦ ↑(f x)) * fun a ↦ ‖g a‖ₑ) a ∂μ < ∞ ↔ ∫⁻ (a : α), ‖↑(f a) • g a‖ₑ ∂μ < ∞
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g : α → E
H : AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ
⊢ AEMeasurable (fun a ↦ ‖g a‖ₑ) (μ.withDensity fun x ↦ ↑(f x))

Tactic:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g : α → E
H : AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ
⊢ AEMeasurable (fun a ↦ ‖g a‖ₑ) (μ.withDensity fun x ↦ ↑(f x))
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g : α → E
H : ¬AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ
⊢ Integrable g (μ.withDensity fun x ↦ ↑(f x)) ↔ Integrable (fun x ↦ ↑(f x) • g x) μ

Tactic:
·

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g : α → E
H : ¬AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ
⊢ Integrable g (μ.withDensity fun x ↦ ↑(f x)) ↔ Integrable (fun x ↦ ↑(f x) • g x) μ
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case pos
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : MultilinearMap 𝕜 E G
C : ℝ
hC : 0 ≤ C
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
m₁ m₂ : (i : ι) → E i
i : ι
s : Finset ι
his : i ∉ s
Hrec : ‖f m₁ - f (s.piecewise m₂ m₁)‖ ≤ C * ∑ i ∈ s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖
A : (insert i s).piecewise m₂ m₁ = update (s.piecewise m₂ m₁) i (m₂ i)
B : s.piecewise m₂ m₁ = update (s.piecewise m₂ m₁) i (m₁ i)
j : ι
a✝ : j ∈ univ
h : j = i
⊢ ‖update (s.piecewise m₂ m₁) i (m₁ i - m₂ i) j‖ ≤ if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖

pp: case neg
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : MultilinearMap 𝕜 E G
C : ℝ
hC : 0 ≤ C
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
m₁ m₂ : (i : ι) → E i
i : ι
s : Finset ι
his : i ∉ s
Hrec : ‖f m₁ - f (s.piecewise m₂ m₁)‖ ≤ C * ∑ i ∈ s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖
A : (insert i s).piecewise m₂ m₁ = update (s.piecewise m₂ m₁) i (m₂ i)
B : s.piecewise m₂ m₁ = update (s.piecewise m₂ m₁) i (m₁ i)
j : ι
a✝ : j ∈ univ
h : ¬j = i
⊢ ‖update (s.piecewise m₂ m₁) i (m₁ i - m₂ i) j‖ ≤ if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖

Tactic:
·

Goal state after:
case pos
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : MultilinearMap 𝕜 E G
C : ℝ
hC : 0 ≤ C
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
m₁ m₂ : (i : ι) → E i
i : ι
s : Finset ι
his : i ∉ s
Hrec : ‖f m₁ - f (s.piecewise m₂ m₁)‖ ≤ C * ∑ i ∈ s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖
A : (insert i s).piecewise m₂ m₁ = update (s.piecewise m₂ m₁) i (m₂ i)
B : s.piecewise m₂ m₁ = update (s.piecewise m₂ m₁) i (m₁ i)
j : ι
a✝ : j ∈ univ
h : j = i
⊢ ‖update (s.piecewise m₂ m₁) i (m₁ i - m₂ i) j‖ ≤ if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0∞
hf : Measurable f
hflt : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → E
⊢ Integrable g (μ.withDensity f) ↔ Integrable (fun x ↦ (f x).toReal • g x) μ

Tactic:
rw [← withDensity_congr_ae (coe_toNNReal_ae_eq hflt),
    integrable_withDensity_iff_integrable_smul]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0∞
hf : Measurable f
hflt : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → E
⊢ Integrable (fun x ↦ (f x).toNNReal • g x) μ ↔ Integrable (fun x ↦ (f x).toReal • g x) μ

case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0∞
hf : Measurable f
hflt : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → E
⊢ Measurable fun x ↦ (f x).toNNReal
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0∞
hf : Measurable f
hflt : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → E
⊢ Integrable (fun x ↦ (f x).toNNReal • g x) μ ↔ Integrable (fun x ↦ (f x).toReal • g x) μ

pp: case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0∞
hf : Measurable f
hflt : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → E
⊢ Measurable fun x ↦ (f x).toNNReal

Tactic:
·

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0∞
hf : Measurable f
hflt : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → E
⊢ Integrable (fun x ↦ (f x).toNNReal • g x) μ ↔ Integrable (fun x ↦ (f x).toReal • g x) μ
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0∞
hf : Measurable f
hflt : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → E
⊢ Measurable fun x ↦ (f x).toNNReal

Tactic:
·

Goal state after:
case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0∞
hf : Measurable f
hflt : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → E
⊢ Measurable fun x ↦ (f x).toNNReal
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case mp
α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
l : Filter α
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
s : Set β
hs : s ∈ map e l ∧ IntegrableOn (f ∘ e) (e ⁻¹' s) μ
⊢ ∃ s ∈ l, IntegrableOn (f ∘ e) s μ

pp: case mpr
α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
l : Filter α
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
s : Set α
hs : s ∈ l ∧ IntegrableOn (f ∘ e) s μ
⊢ ∃ s ∈ map e l, IntegrableOn (f ∘ e) (e ⁻¹' s) μ

Tactic:
·

Goal state after:
case mp
α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
l : Filter α
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
s : Set β
hs : s ∈ map e l ∧ IntegrableOn (f ∘ e) (e ⁻¹' s) μ
⊢ ∃ s ∈ l, IntegrableOn (f ∘ e) s μ
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case neg
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : MultilinearMap 𝕜 E G
C : ℝ
hC : 0 ≤ C
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
m₁ m₂ : (i : ι) → E i
i : ι
s : Finset ι
his : i ∉ s
Hrec : ‖f m₁ - f (s.piecewise m₂ m₁)‖ ≤ C * ∑ i ∈ s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖
A : (insert i s).piecewise m₂ m₁ = update (s.piecewise m₂ m₁) i (m₂ i)
B : s.piecewise m₂ m₁ = update (s.piecewise m₂ m₁) i (m₁ i)
j : ι
a✝ : j ∈ univ
h : ¬j = i
⊢ ‖update (s.piecewise m₂ m₁) i (m₁ i - m₂ i) j‖ ≤ if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖

Tactic:
·

Goal state after:
case neg
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : MultilinearMap 𝕜 E G
C : ℝ
hC : 0 ≤ C
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
m₁ m₂ : (i : ι) → E i
i : ι
s : Finset ι
his : i ∉ s
Hrec : ‖f m₁ - f (s.piecewise m₂ m₁)‖ ≤ C * ∑ i ∈ s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖
A : (insert i s).piecewise m₂ m₁ = update (s.piecewise m₂ m₁) i (m₂ i)
B : s.piecewise m₂ m₁ = update (s.piecewise m₂ m₁) i (m₁ i)
j : ι
a✝ : j ∈ univ
h : ¬j = i
⊢ ‖update (s.piecewise m₂ m₁) i (m₁ i - m₂ i) j‖ ≤ if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case mpr
α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
l : Filter α
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
s : Set α
hs : s ∈ l ∧ IntegrableOn (f ∘ e) s μ
⊢ ∃ s ∈ map e l, IntegrableOn (f ∘ e) (e ⁻¹' s) μ

Tactic:
·

Goal state after:
case mpr
α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
l : Filter α
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
s : Set α
hs : s ∈ l ∧ IntegrableOn (f ∘ e) s μ
⊢ ∃ s ∈ map e l, IntegrableOn (f ∘ e) (e ⁻¹' s) μ
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case mp
α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
l : Filter α
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
μ : Measure β
s : Set β
hs : s ∈ map e l
int : IntegrableOn f s μ
⊢ ∃ s ∈ map e l, IntegrableOn f s (μ.restrict (range e))

pp: case mpr
α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
l : Filter α
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
μ : Measure β
s : Set β
hs : s ∈ map e l
int : IntegrableOn f s (μ.restrict (range e))
⊢ ∃ s ∈ map e l, IntegrableOn f s μ

Tactic:
·

Goal state after:
case mp
α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
l : Filter α
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
μ : Measure β
s : Set β
hs : s ∈ map e l
int : IntegrableOn f s μ
⊢ ∃ s ∈ map e l, IntegrableOn f s (μ.restrict (range e))
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case mpr
α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
l : Filter α
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
μ : Measure β
s : Set β
hs : s ∈ map e l
int : IntegrableOn f s (μ.restrict (range e))
⊢ ∃ s ∈ map e l, IntegrableOn f s μ

Tactic:
·

Goal state after:
case mpr
α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
l : Filter α
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
μ : Measure β
s : Set β
hs : s ∈ map e l
int : IntegrableOn f s (μ.restrict (range e))
⊢ ∃ s ∈ map e l, IntegrableOn f s μ