mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
μ : Measure α
f : α → ℝ
hf : Measurable f
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
hfi : Integrable f μ
⊢ s.HaveLebesgueDecomposition μ

pp: case neg
α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
μ : Measure α
f : α → ℝ
hf : Measurable f
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
hfi : ¬Integrable f μ
⊢ s.HaveLebesgueDecomposition μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
μ : Measure α
f : α → ℝ
hf : Measurable f
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
hfi : Integrable f μ
⊢ s.HaveLebesgueDecomposition μ
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
μ : Measure α
f : α → ℝ
hf : Measurable f
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
hfi : ¬Integrable f μ
⊢ s.HaveLebesgueDecomposition μ

Tactic:
·

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
μ : Measure α
f : α → ℝ
hf : Measurable f
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
hfi : ¬Integrable f μ
⊢ s.HaveLebesgueDecomposition μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case inl
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : Module ℚ≥0 E
inst✝ : NormedSpace ℝ E
ι : Type u_3
f : ι → α → E
hp : 1 ≤ p
hf : ∀ i ∈ ∅, MemLp (f i) p μ
⊢ lpNorm (𝔼 i ∈ ∅, f i) p μ ≤ 𝔼 i ∈ ∅, lpNorm (f i) p μ

pp: case inr
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : Module ℚ≥0 E
inst✝ : NormedSpace ℝ E
ι : Type u_3
s : Finset ι
f : ι → α → E
hf : ∀ i ∈ s, MemLp (f i) p μ
hp : 1 ≤ p
hs : s.Nonempty
⊢ lpNorm (𝔼 i ∈ s, f i) p μ ≤ 𝔼 i ∈ s, lpNorm (f i) p μ

Tactic:
·

Goal state after:
case inl
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : Module ℚ≥0 E
inst✝ : NormedSpace ℝ E
ι : Type u_3
f : ι → α → E
hp : 1 ≤ p
hf : ∀ i ∈ ∅, MemLp (f i) p μ
⊢ lpNorm (𝔼 i ∈ ∅, f i) p μ ≤ 𝔼 i ∈ ∅, lpNorm (f i) p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
g : α → ℝ
hg : MemLp g p μ
h : ∀ (x : α), ‖f x‖ ≤ g x
hf : AEStronglyMeasurable f μ
⊢ lpNorm f p μ ≤ lpNorm g p μ

pp: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
g : α → ℝ
hg : MemLp g p μ
h : ∀ (x : α), ‖f x‖ ≤ g x
hf : ¬AEStronglyMeasurable f μ
⊢ lpNorm f p μ ≤ lpNorm g p μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
g : α → ℝ
hg : MemLp g p μ
h : ∀ (x : α), ‖f x‖ ≤ g x
hf : AEStronglyMeasurable f μ
⊢ lpNorm f p μ ≤ lpNorm g p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
g : α → ℝ
hg : MemLp g p μ
h : ∀ (x : α), ‖f x‖ ≤ g x
hf : AEStronglyMeasurable f μ
⊢ lpNorm f p μ ≤ lpNorm g p μ

Tactic:
rw [← toReal_eLpNorm hf, ← toReal_eLpNorm hg.aestronglyMeasurable]
    exact ENNReal.toNNReal_mono (hg.eLpNorm_ne_top) (eLpNorm_mono_real h)

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
g : α → ℝ
hg : MemLp g p μ
h : ∀ (x : α), ‖f x‖ ≤ g x
hf : ¬AEStronglyMeasurable f μ
⊢ lpNorm f p μ ≤ lpNorm g p μ

Tactic:
·

Goal state after:
case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
g : α → ℝ
hg : MemLp g p μ
h : ∀ (x : α), ‖f x‖ ≤ g x
hf : ¬AEStronglyMeasurable f μ
⊢ lpNorm f p μ ≤ lpNorm g p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
c : ℝ≥0
hc : c ≠ 0
hf : AEStronglyMeasurable f μ
⊢ lpNorm f p (c • μ) = c ^ p.toReal⁻¹ • lpNorm f p μ

pp: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
c : ℝ≥0
hc : c ≠ 0
hf : ¬AEStronglyMeasurable f μ
⊢ lpNorm f p (c • μ) = c ^ p.toReal⁻¹ • lpNorm f p μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
c : ℝ≥0
hc : c ≠ 0
hf : AEStronglyMeasurable f μ
⊢ lpNorm f p (c • μ) = c ^ p.toReal⁻¹ • lpNorm f p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
c : ℝ≥0
hc : c ≠ 0
hf : ¬AEStronglyMeasurable f μ
⊢ lpNorm f p (c • μ) = c ^ p.toReal⁻¹ • lpNorm f p μ

Tactic:
·

Goal state after:
case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
c : ℝ≥0
hc : c ≠ 0
hf : ¬AEStronglyMeasurable f μ
⊢ lpNorm f p (c • μ) = c ^ p.toReal⁻¹ • lpNorm f p 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
p : ℝ≥0∞
f : ℕ → α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hf_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n) p μ ≤ ε
ε : ℝ
hε : 0 < ε
N : ℕ
hN : ∀ n ≥ N, eLpNorm (f n) p μ ≤ ENNReal.ofReal ε
F : Fin N → α → β := fun n ↦ f ↑n
hF : ∀ (n : Fin N), MemLp (F n) p μ
δ₁ : ℝ
hδpos₁ : 0 < δ₁
hδ₁ :
  ∀ (i : Fin N) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (F i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal δ₁
hn : n < N
⊢ eLpNorm (s.indicator (f n)) p μ ≤ ENNReal.ofReal ε

pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hf_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n) p μ ≤ ε
ε : ℝ
hε : 0 < ε
N : ℕ
hN : ∀ n ≥ N, eLpNorm (f n) p μ ≤ ENNReal.ofReal ε
F : Fin N → α → β := fun n ↦ f ↑n
hF : ∀ (n : Fin N), MemLp (F n) p μ
δ₁ : ℝ
hδpos₁ : 0 < δ₁
hδ₁ :
  ∀ (i : Fin N) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (F i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal δ₁
hn : N ≤ n
⊢ eLpNorm (s.indicator (f n)) p μ ≤ ENNReal.ofReal ε

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hf_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n) p μ ≤ ε
ε : ℝ
hε : 0 < ε
N : ℕ
hN : ∀ n ≥ N, eLpNorm (f n) p μ ≤ ENNReal.ofReal ε
F : Fin N → α → β := fun n ↦ f ↑n
hF : ∀ (n : Fin N), MemLp (F n) p μ
δ₁ : ℝ
hδpos₁ : 0 < δ₁
hδ₁ :
  ∀ (i : Fin N) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (F i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal δ₁
hn : n < N
⊢ eLpNorm (s.indicator (f n)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : p ≠ ∞
f : α → E
c : ℝ≥0
hf : AEStronglyMeasurable f μ
⊢ lpNorm f p (c • μ) = c ^ p.toReal⁻¹ • lpNorm f p μ

pp: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : p ≠ ∞
f : α → E
c : ℝ≥0
hf : ¬AEStronglyMeasurable f μ
⊢ lpNorm f p (c • μ) = c ^ p.toReal⁻¹ • lpNorm f p μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : p ≠ ∞
f : α → E
c : ℝ≥0
hf : AEStronglyMeasurable f μ
⊢ lpNorm f p (c • μ) = c ^ p.toReal⁻¹ • lpNorm f p μ
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: case e_a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : SignedMeasure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
htμ : t.toJordanDecomposition.posPart ⟂ₘ μ ∧ t.toJordanDecomposition.negPart ⟂ₘ μ
hadd : s = t + μ.withDensityᵥ f
htμ' : t ⟂ᵥ μ.toENNRealVectorMeasure
⊢ t.toJordanDecomposition.posPart = s.toJordanDecomposition.posPart.singularPart μ

pp: case e_a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : SignedMeasure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
htμ : t.toJordanDecomposition.posPart ⟂ₘ μ ∧ t.toJordanDecomposition.negPart ⟂ₘ μ
hadd : s = t + μ.withDensityᵥ f
htμ' : t ⟂ᵥ μ.toENNRealVectorMeasure
⊢ t.toJordanDecomposition.negPart = s.toJordanDecomposition.negPart.singularPart μ

Tactic:
·

Goal state after:
case e_a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : SignedMeasure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
htμ : t.toJordanDecomposition.posPart ⟂ₘ μ ∧ t.toJordanDecomposition.negPart ⟂ₘ μ
hadd : s = t + μ.withDensityᵥ f
htμ' : t ⟂ᵥ μ.toENNRealVectorMeasure
⊢ t.toJordanDecomposition.posPart = s.toJordanDecomposition.posPart.singularPart μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case neg.inl
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
c : ℝ≥0
hf : ¬AEStronglyMeasurable f μ
hp : 0 ≠ ∞
⊢ lpNorm f 0 (c • μ) = c ^ (ENNReal.toReal 0)⁻¹ • lpNorm f 0 μ

pp: case neg.inr
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : p ≠ ∞
f : α → E
c : ℝ≥0
hf : ¬AEStronglyMeasurable f μ
hp₀ : p ≠ 0
⊢ lpNorm f p (c • μ) = c ^ p.toReal⁻¹ • lpNorm f p μ

Tactic:
·

Goal state after:
case neg.inl
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
c : ℝ≥0
hf : ¬AEStronglyMeasurable f μ
hp : 0 ≠ ∞
⊢ lpNorm f 0 (c • μ) = c ^ (ENNReal.toReal 0)⁻¹ • lpNorm f 0 μ
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: case e_a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : SignedMeasure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
htμ : t.toJordanDecomposition.posPart ⟂ₘ μ ∧ t.toJordanDecomposition.negPart ⟂ₘ μ
hadd : s = t + μ.withDensityᵥ f
htμ' : t ⟂ᵥ μ.toENNRealVectorMeasure
⊢ t.toJordanDecomposition.negPart = s.toJordanDecomposition.negPart.singularPart μ

Tactic:
·

Goal state after:
case e_a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : SignedMeasure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
htμ : t.toJordanDecomposition.posPart ⟂ₘ μ ∧ t.toJordanDecomposition.negPart ⟂ₘ μ
hadd : s = t + μ.withDensityᵥ f
htμ' : t ⟂ᵥ μ.toENNRealVectorMeasure
⊢ t.toJordanDecomposition.negPart = s.toJordanDecomposition.negPart.singularPart μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case neg.inr.inl
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : p ≠ ∞
f : α → E
hf : ¬AEStronglyMeasurable f μ
hp₀ : p ≠ 0
⊢ lpNorm f p (0 • μ) = 0 ^ p.toReal⁻¹ • lpNorm f p μ

pp: case neg.inr.inr
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : p ≠ ∞
f : α → E
c : ℝ≥0
hf : ¬AEStronglyMeasurable f μ
hp₀ : p ≠ 0
hc : c ≠ 0
⊢ lpNorm f p (c • μ) = c ^ p.toReal⁻¹ • lpNorm f p μ

Tactic:
·

Goal state after:
case neg.inr.inl
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : p ≠ ∞
f : α → E
hf : ¬AEStronglyMeasurable f μ
hp₀ : p ≠ 0
⊢ lpNorm f p (0 • μ) = 0 ^ p.toReal⁻¹ • lpNorm f p μ
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : SignedMeasure α
f : α → ℝ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
hfi : Integrable f μ
⊢ t = s.singularPart μ

pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : SignedMeasure α
f : α → ℝ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
hfi : ¬Integrable f μ
⊢ t = s.singularPart μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : SignedMeasure α
f : α → ℝ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
hfi : Integrable f μ
⊢ t = s.singularPart μ
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : SignedMeasure α
f : α → ℝ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
hfi : ¬Integrable f μ
⊢ t = s.singularPart μ

Tactic:
·

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : SignedMeasure α
f : α → ℝ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
hfi : ¬Integrable f μ
⊢ t = s.singularPart μ
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hf_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n) p μ ≤ ε
ε : ℝ
hε : 0 < ε
N : ℕ
hN : ∀ n ≥ N, eLpNorm (f n) p μ ≤ ENNReal.ofReal ε
F : Fin N → α → β := fun n ↦ f ↑n
hF : ∀ (n : Fin N), MemLp (F n) p μ
δ₁ : ℝ
hδpos₁ : 0 < δ₁
hδ₁ :
  ∀ (i : Fin N) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (F i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal δ₁
hn : N ≤ n
⊢ eLpNorm (s.indicator (f n)) p μ ≤ ENNReal.ofReal ε

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hf_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n) p μ ≤ ε
ε : ℝ
hε : 0 < ε
N : ℕ
hN : ∀ n ≥ N, eLpNorm (f n) p μ ≤ ENNReal.ofReal ε
F : Fin N → α → β := fun n ↦ f ↑n
hF : ∀ (n : Fin N), MemLp (F n) p μ
δ₁ : ℝ
hδpos₁ : 0 < δ₁
hδ₁ :
  ∀ (i : Fin N) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (F i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal δ₁
hn : N ≤ n
⊢ eLpNorm (s.indicator (f n)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
K : Type u_3
inst✝ : RCLike K
f : α → K
p : ℝ≥0∞
μ : Measure α
hf : AEStronglyMeasurable f μ
⊢ lpNorm ((starRingEnd (α → K)) f) p μ = lpNorm f p μ

pp: case neg
α : Type u_1
m : MeasurableSpace α
K : Type u_3
inst✝ : RCLike K
f : α → K
p : ℝ≥0∞
μ : Measure α
hf : ¬AEStronglyMeasurable f μ
⊢ lpNorm ((starRingEnd (α → K)) f) p μ = lpNorm f p μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
K : Type u_3
inst✝ : RCLike K
f : α → K
p : ℝ≥0∞
μ : Measure α
hf : AEStronglyMeasurable f μ
⊢ lpNorm ((starRingEnd (α → K)) f) p μ = lpNorm f p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
K : Type u_3
inst✝ : RCLike K
f : α → K
p : ℝ≥0∞
μ : Measure α
hf : AEStronglyMeasurable f μ
⊢ lpNorm (fun x ↦ ‖(starRingEnd (α → K)) f x‖) p μ = lpNorm (fun x ↦ ‖f x‖) p μ

pp: case pos.hf
α : Type u_1
m : MeasurableSpace α
K : Type u_3
inst✝ : RCLike K
f : α → K
p : ℝ≥0∞
μ : Measure α
hf : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable ((starRingEnd (α → K)) f) μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
K : Type u_3
inst✝ : RCLike K
f : α → K
p : ℝ≥0∞
μ : Measure α
hf : AEStronglyMeasurable f μ
⊢ lpNorm (fun x ↦ ‖(starRingEnd (α → K)) f x‖) p μ = lpNorm (fun x ↦ ‖f x‖) p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case pos.hf
α : Type u_1
m : MeasurableSpace α
K : Type u_3
inst✝ : RCLike K
f : α → K
p : ℝ≥0∞
μ : Measure α
hf : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable ((starRingEnd (α → K)) f) μ

Tactic:
·

Goal state after:
case pos.hf
α : Type u_1
m : MeasurableSpace α
K : Type u_3
inst✝ : RCLike K
f : α → K
p : ℝ≥0∞
μ : Measure α
hf : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable ((starRingEnd (α → K)) f) μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
K : Type u_3
inst✝ : RCLike K
f : α → K
p : ℝ≥0∞
μ : Measure α
hf : ¬AEStronglyMeasurable f μ
⊢ lpNorm ((starRingEnd (α → K)) f) p μ = lpNorm f p μ

Tactic:
·

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
K : Type u_3
inst✝ : RCLike K
f : α → K
p : ℝ≥0∞
μ : Measure α
hf : ¬AEStronglyMeasurable f μ
⊢ lpNorm ((starRingEnd (α → K)) f) p μ = lpNorm f p μ
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hg : MemLp g p μ
hfg : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0)
this : f = (fun x ↦ g) + fun n ↦ f n - g
⊢ UnifIntegrable (fun x ↦ g) p μ

pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hg : MemLp g p μ
hfg : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0)
this : f = (fun x ↦ g) + fun n ↦ f n - g
⊢ UnifIntegrable (fun n ↦ f n - g) p μ

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hg : MemLp g p μ
hfg : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0)
this : f = (fun x ↦ g) + fun n ↦ f n - g
⊢ UnifIntegrable (fun x ↦ g) p μ
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hg : MemLp g p μ
hfg : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0)
this : f = (fun x ↦ g) + fun n ↦ f n - g
⊢ UnifIntegrable (fun n ↦ f n - g) p μ

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hg : MemLp g p μ
hfg : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0)
this : f = (fun x ↦ g) + fun n ↦ f n - g
⊢ UnifIntegrable (fun n ↦ f n - g) p μ
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosed

Goal state before:
pp: case refine_1
Ω : Type u_1
inst✝⁴ : TopologicalSpace Ω
inst✝³ : MeasurableSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝¹ : L.IsCountablyGenerated
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
fs : ι → Ω →ᵇ ℝ≥0
c : ℝ≥0
fs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
f : Ω → ℝ≥0
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i ↦ (fs i) ω) L (𝓝 (f ω))
⊢ ∀ᶠ (n : ι) in L, ∀ᵐ (a : Ω) ∂μ, ↑((fs n) a) ≤ ↑c

pp: case refine_2
Ω : Type u_1
inst✝⁴ : TopologicalSpace Ω
inst✝³ : MeasurableSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝¹ : L.IsCountablyGenerated
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
fs : ι → Ω →ᵇ ℝ≥0
c : ℝ≥0
fs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
f : Ω → ℝ≥0
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i ↦ (fs i) ω) L (𝓝 (f ω))
⊢ ∀ᵐ (a : Ω) ∂μ, Tendsto (fun n ↦ ↑((fs n) a)) L (𝓝 ↑(f a))

Tactic:
·

Goal state after:
case refine_1
Ω : Type u_1
inst✝⁴ : TopologicalSpace Ω
inst✝³ : MeasurableSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝¹ : L.IsCountablyGenerated
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
fs : ι → Ω →ᵇ ℝ≥0
c : ℝ≥0
fs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
f : Ω → ℝ≥0
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i ↦ (fs i) ω) L (𝓝 (f ω))
⊢ ∀ᶠ (n : ι) in L, ∀ᵐ (a : Ω) ∂μ, ↑((fs n) a) ≤ ↑c
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosed

Goal state before:
pp: case refine_2
Ω : Type u_1
inst✝⁴ : TopologicalSpace Ω
inst✝³ : MeasurableSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝¹ : L.IsCountablyGenerated
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
fs : ι → Ω →ᵇ ℝ≥0
c : ℝ≥0
fs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
f : Ω → ℝ≥0
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i ↦ (fs i) ω) L (𝓝 (f ω))
⊢ ∀ᵐ (a : Ω) ∂μ, Tendsto (fun n ↦ ↑((fs n) a)) L (𝓝 ↑(f a))

Tactic:
·

Goal state after:
case refine_2
Ω : Type u_1
inst✝⁴ : TopologicalSpace Ω
inst✝³ : MeasurableSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝¹ : L.IsCountablyGenerated
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
fs : ι → Ω →ᵇ ℝ≥0
c : ℝ≥0
fs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
f : Ω → ℝ≥0
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i ↦ (fs i) ω) L (𝓝 (f ω))
⊢ ∀ᵐ (a : Ω) ∂μ, Tendsto (fun n ↦ ↑((fs n) a)) L (𝓝 ↑(f a))
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : μ univ = 0
⊢ UnifIntegrable f p μ

pp: case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
⊢ UnifIntegrable f p μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : μ univ = 0
⊢ UnifIntegrable f p μ