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

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
β : Type u_3
inst✝ : MeasurableSpace β
μb : Measure β
f : α → β
s : Set β
hs : MeasurableSet s
hμs : μb s ≠ ∞
c : E
hf : MeasurePreserving f μ μb
⊢ μ (f ⁻¹' s) ≠ ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
inst✝ : IsFiniteMeasure μ
f : α → β
hf : AEStronglyMeasurable f μ
p q : ℝ
hp✝ : 0 ≤ p
hq✝ : 0 ≤ q
hpq : p ≤ q
hint : Integrable (fun x ↦ ‖f x‖ ^ (ENNReal.ofReal q).toReal) μ
hp : 0 < p
hq : 0 < q
⊢ ENNReal.ofReal q ≠ 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
inst✝ : IsFiniteMeasure μ
f : α → β
hf : AEStronglyMeasurable f μ
p q : ℝ
hp✝ : 0 ≤ p
hq✝ : 0 ≤ q
hpq : p ≤ q
hint : Integrable (fun x ↦ ‖f x‖ ^ (ENNReal.ofReal q).toReal) μ
hp : 0 < p
hq : 0 < q
⊢ ENNReal.ofReal q ≠ ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
inst✝ : IsFiniteMeasure μ
f : α → β
hf : AEStronglyMeasurable f μ
p q : ℝ
hp : 0 ≤ p
hq : 0 ≤ q
hpq : p ≤ q
hint : Integrable (fun x ↦ ‖f x‖ ^ q) μ
⊢ Integrable (fun x ↦ ‖f x‖ ^ p) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
inst✝ : IsFiniteMeasure μ
f : α → β
hf : AEStronglyMeasurable f μ
p q : ℕ
hpq : p ≤ q
hint : Integrable (fun x ↦ ‖f x‖ ^ ↑q) μ
⊢ ↑p ≤ ↑q

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
inst✝ : IsFiniteMeasure μ
f : α → β
hf : AEStronglyMeasurable f μ
p q : ℕ
hpq : p ≤ q
hint : Integrable (fun x ↦ ‖f x‖ ^ q) μ
⊢ Integrable (fun x ↦ ‖f x‖ ^ p) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_8
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
μ' : Measure α
c : ℝ≥0∞
hc : c ≠ ∞
hμ'_le : μ' ≤ c • μ
f : α → ε
hf : Integrable f μ
⊢ Integrable f μ'

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
s : Set α
inst✝ : NormedSpace ℝ E
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E
⊢ indicatorConstLp 2 hs hμs x = (ContinuousLinearMap.toSpanSingleton ℝ x).compLp (indicatorConstLp 2 hs hμs 1)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : PseudoMetrizableSpace ε
f : α → ε
hμ : Integrable f μ
hν : Integrable f ν
⊢ Integrable f (μ + ν)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
hf₀ : ∀ {m : (i : ι) → E i} {i : ι}, ‖m i‖ = 0 → ‖f m‖ = 0
ε : ι → ℝ
C : ℝ
hε : ∀ (i : ι), 0 < ε i
c : ι → 𝕜
hc : ∀ (i : ι), 1 < ‖c i‖
hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖
m : (i : ι) → E i
⊢ ‖f m‖ ≤ C * ∏ i, ‖m i‖

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝ : TopologicalSpace β
μ : Measure α
f : α → β
⊢ AEStronglyMeasurable f (μ.restrict ∅)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝ : TopologicalSpace β
l : Filter α
f : α → β
μ : Measure α
h : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable f (μ.restrict univ)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝ : NormedAddCommGroup E
f : α → E
s : Set α
μ : Measure α
C : ℝ
hs : μ s < ∞
hf : ∀ᵐ (x : α) ∂μ.restrict s, ‖f x‖ ≤ C
⊢ (μ.restrict s) univ < ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_3
mα : MeasurableSpace α
f : α → ε
s : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
C : ℝ≥0∞
hC : ‖C‖ₑ ≠ ∞
hs : μ s ≠ ∞
hf : ∀ᵐ (x : α) ∂μ.restrict s, ‖f x‖ₑ ≤ C
⊢ (μ.restrict s) univ < ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_3
mα : MeasurableSpace α
f : α → ε
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
⊢ IntegrableOn f ∅ μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_3
mα : MeasurableSpace α
f : α → ε
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
⊢ IntegrableOn f univ μ ↔ Integrable f μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_3
mα : MeasurableSpace α
f : α → ε
s : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
hs : μ s = 0
⊢ IntegrableOn f s μ

Tactic:
by

Goal state after:
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'
C : ε'
hC : ‖C‖ₑ ≠ ∞
⊢ IntegrableOn (fun x ↦ C) s μ ↔ ‖C‖ₑ = 0 ∨ μ s < ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
h : Integrable f (μ + ν)
⊢ Integrable f μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
h : Integrable f (μ + ν)
⊢ Integrable f ν

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
⊢ Integrable f 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_3
mα : MeasurableSpace α
f : α → ε
s t : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
h : IntegrableOn f s μ
⊢ IntegrableOn f s (μ.restrict t)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : MeasurableSingletonClass α
a : α
f : α → ε
hfa : ‖f a‖ₑ < ∞
⊢ HasFiniteIntegral f (Measure.dirac a)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
a : α
f : α → ε
hf : StronglyMeasurable f
hfa : ‖f a‖ₑ < ∞
⊢ HasFiniteIntegral f (Measure.dirac a)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_3
mα : MeasurableSpace α
f : α → ε
s t : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
h : IntegrableOn f s (μ.restrict t)
⊢ IntegrableOn f (s ∩ t) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_5
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : PseudoMetrizableSpace ε
ι : Type u_8
m : MeasurableSpace α
f : α → ε
μ : ι → Measure α
s : Finset ι
⊢ Integrable f (∑ i ∈ s, μ i) ↔ ∀ i ∈ s, Integrable f (μ i)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_8
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
h : Integrable f μ
c : ℝ≥0∞
hc : c ≠ ∞
⊢ Integrable f (c • μ)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_8
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
h : Integrable f μ
c : ℝ≥0
⊢ Integrable f (c • μ)

Tactic:
by

Goal state after:
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✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
f g : α → ε'
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : IntegrableOn f s μ
hg : IntegrableOn g sᶜ μ
⊢ Integrable (s.piecewise f g) μ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
ξ₀ : 𝕜
hξ₀ : 0 < ‖ξ₀‖
hξ₀ε : ‖ξ₀‖ < ε
this✝¹ : {ξ₀}ᶜ ∈ 𝓝 0
this✝ : balancedCore 𝕜 {ξ₀}ᶜ ∈ 𝓝 0
ξ : 𝕜
hξ : ξ ∈ balancedCore 𝕜 {ξ₀}ᶜ
hξ0 : ¬ξ = 0
h : ε < ‖ξ‖
this : (ξ₀ * ξ⁻¹) • ξ ∈ balancedCore 𝕜 {ξ₀}ᶜ
⊢ False

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_8
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
c : ℝ≥0∞
h₁ : c ≠ 0
h₂ : c ≠ ∞
h : Integrable f (c • μ)
⊢ Integrable f μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
f : α → ε'
x : α
inst✝ : MeasurableSingletonClass α
hfx : ‖f x‖ₑ ≠ ∞
⊢ f =ᵐ[μ.restrict {x}] fun x_1 ↦ f x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_8
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
c : ℝ≥0∞
h₁ : c ≠ 0
h₂ : c ≠ ∞
⊢ c⁻¹ ≠ 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_8
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
c : ℝ≥0∞
h₁ : c ≠ 0
h₂ : c ≠ ∞
⊢ c⁻¹ ≠ ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_8
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
h : Integrable f μ
⊢ Integrable f ((μ univ)⁻¹ • μ)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_8
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
inst✝ : IsFiniteMeasure μ
f : α → ε
h : μ = 0
⊢ Integrable f ((μ univ)⁻¹ • μ) ↔ Integrable f μ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
⊢ map id (𝓝 0) = map (fun x ↦ id x • 1) (𝓝 0)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_8
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
inst✝ : IsFiniteMeasure μ
f : α → ε
h : μ ≠ 0
⊢ μ univ ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
⊢ 𝓝 (0 • 1) = 𝓝 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:

Tactic:
@nhds 𝕜 hnorm.toUniformSpace.toTopologicalSpace 0 =
          map id (@nhds 𝕜 hnorm.toUniformSpace.toTopologicalSpace 0) :=
        map_id.symm
      _ = map (fun x => id x • (1 : 𝕜)) (@nhds 𝕜 hnorm.toUniformSpace.toTopologicalSpace 0) := by
        simp
      _ ≤ @nhds 𝕜 t ((0 : 𝕜) • (1 : 𝕜)) :=
        (@Tendsto.smul_const _ _ _ hnorm.toUniformSpace.toTopologicalSpace t _ _ _ _ _
          tendsto_id (1 : 𝕜))
      _ = @nhds 𝕜 t 0 := by rw [zero_smul]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
⊢ t = PseudoMetricSpace.toUniformSpace.toTopologicalSpace

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_8
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
inst✝ : IsFiniteMeasure μ
f : α → ε
⊢ Integrable f ((μ univ)⁻¹ • μ) ↔ Integrable f μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
f : α → ε'
x : α
inst✝ : MeasurableSingletonClass α
hfx : ‖f x‖ₑ ≠ ∞
⊢ IntegrableOn f {x} μ ↔ ‖f x‖ₑ = 0 ∨ μ {x} < ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
α' : Type u_8
inst✝ : MeasurableSpace α'
f : α → α'
g : α' → ε
hg : AEStronglyMeasurable g (Measure.map f μ)
hf : AEMeasurable f μ
⊢ Integrable g (Measure.map f μ) ↔ Integrable (g ∘ f) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
δ : Type u_4
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → δ
hf : MeasurableEmbedding f
g : δ → ε
⊢ Integrable g (Measure.map f μ) ↔ Integrable (g ∘ f) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
f : α → ε
μ : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : PseudoMetrizableSpace ε
s : Set β
hs : s.Finite
t : β → Set α
⊢ IntegrableOn f (⋃ i ∈ s, t i) μ ↔ ∀ i ∈ s, IntegrableOn f (t i) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
f : α → ε
μ : Measure α
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
inst✝¹ : PseudoMetrizableSpace ε
inst✝ : Finite β
t : β → Set α
⊢ IntegrableOn f (⋃ i, t i) μ ↔ ∀ (i : β), IntegrableOn f (t i) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
δ : Type u_4
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α ≃ᵐ δ
g : δ → ε
⊢ Integrable g (Measure.map (⇑f) μ) ↔ Integrable (g ∘ ⇑f) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSingletonClass α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset α
f : α → E
⊢ IntegrableOn f (↑s) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSingletonClass α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Set α
hs : s.Finite
f : α → E
⊢ IntegrableOn f s μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
δ : Type u_4
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
ν : Measure δ
g : δ → ε
f : α → δ
hf : MeasurePreserving f μ ν
hg : AEStronglyMeasurable g ν
⊢ Integrable (g ∘ f) μ ↔ Integrable g ν

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_3
mα : MeasurableSpace α
f : α → ε
s : Set α
μ ν : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : PseudoMetrizableSpace ε
hμ : IntegrableOn f s μ
hν : IntegrableOn f s ν
⊢ IntegrableOn f s (μ + ν)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : α → β
hf : Integrable f μ
hg : Integrable g μ
⊢ ∫⁻ (a : α), edist (f a) (0 a) ∂μ < ∞ ∧ ∫⁻ (a : α), edist (g a) (0 a) ∂μ < ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ε' : Type u_8
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
μ : Measure α
⊢ Integrable 0 μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε' : Type u_8
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddCommMonoid ε'
inst✝ : ContinuousAdd ε'
ι : Type u_9
s : Finset ι
f : ι → α → ε'
hf : ∀ i ∈ s, Integrable (f i) μ
⊢ Integrable (fun a ↦ ∑ i ∈ s, f i a) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
μ : Measure α
s : Set β
⊢ IntegrableOn f s (Measure.map e μ) ↔ IntegrableOn (f ∘ e) (e ⁻¹' s) μ

Tactic:
by

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

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 : Integrable f μ
⊢ HasFiniteIntegral (-f) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : α → β
hf : Integrable f μ
⊢ Integrable (g + f) μ ↔ Integrable g μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
μ : Measure β
s : Set β
hs : s ⊆ range e
⊢ IntegrableOn f s μ ↔ IntegrableOn (f ∘ e) (e ⁻¹' s) (Measure.comap e μ)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul 𝕜 E
l : E →ₗ[𝕜] 𝕜
hl : IsClosed[inst✝²] ↑l.ker
H : ¬finrank 𝕜 ↥l.range = 0
this : finrank 𝕜 ↥l.range = 1
⊢ Function.Injective ⇑(l.ker.liftQ l ⋯)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
μ : Measure β
⊢ IntegrableOn f (range e) μ ↔ Integrable (f ∘ e) (Measure.comap e μ)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ
h_meas : AEStronglyMeasurable f μ
hf : 0 ≤ᵐ[μ] f
hg : 0 ≤ᵐ[μ] g
h_int : Integrable (f + g) μ
⊢ Integrable f μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_3
mα : MeasurableSpace α
f : α → ε
s : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
hs : MeasurableSet s
⊢ IntegrableOn f s μ ↔ Integrable (f ∘ Subtype.val) (Measure.comap Subtype.val μ)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : MeasurableSpace β
e : α ≃ᵐ β
f : β → ε
μ : Measure α
s : Set β
⊢ IntegrableOn f s (Measure.map (⇑e) μ) ↔ IntegrableOn (f ∘ ⇑e) (⇑e ⁻¹' s) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ
h_meas : AEStronglyMeasurable f μ
hf : f ≤ᵐ[μ] 0
hg : g ≤ᵐ[μ] 0
⊢ Integrable (f + g) μ ↔ Integrable f μ ∧ Integrable g μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : α → β
hf : Integrable f μ
hg : Integrable g μ
⊢ Integrable (f - g) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : α → β
hf : Integrable f μ
hg : Integrable g μ
⊢ Integrable (fun a ↦ f a - g a) μ

Tactic:
by

Goal state after:
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'
hs : MeasurableSet s
⊢ Integrable (s.indicator f) μ ↔ IntegrableOn f s μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hf : Integrable f μ
⊢ Integrable (fun x ↦ ‖f x‖ₑ) μ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul 𝕜 E
l : E →ₗ[𝕜] 𝕜
hl : IsClosed[inst✝²] ↑l.ker
H : ¬finrank 𝕜 ↥l.range = 0
this : finrank 𝕜 ↥l.range = 1
hi : Function.Injective ⇑(l.ker.liftQ l ⋯)
⊢ Function.Surjective ⇑(l.ker.liftQ l ⋯)

Tactic:
by

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

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 : Integrable f μ
⊢ Integrable (fun a ↦ ‖f a‖) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_7
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
⊢ Integrable (↑↑(indicatorConstLp p hs hμs c)) μ

Tactic:
by

Goal state after: