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

Goal state before:
pp: α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
g : E → F
c : ℝ≥0
hg : LipschitzWith c g
g0 : g 0 = 0
a✝ : α
ha : ↑↑0 a✝ = 0 a✝
⊢ (g ∘ ↑↑0) a✝ = 0 a✝

Tactic:
simp only [ha, g0, Function.comp_apply, Pi.zero_apply]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ p⁻¹ ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
h_tsum_lt_top : (∑' (i : ℕ), B i) ^ p < ∞
⊢ 0 < p⁻¹

Tactic:
simp [hp_pos]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
h_integral : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ < ∞
rpow_ae_lt_top : ∀ᵐ (x : α) ∂μ, (∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ) ^ p < ∞
x : α
hx : ∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ < ∞ ^ p⁻¹
⊢ 0 < p⁻¹

Tactic:
simp [hp_pos]

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

Goal state before:
pp: 𝕜₁ : Type u_2
𝕜₂ : Type u_3
𝕜₃ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_8
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedAddCommGroup G
inst✝¹⁰ : NontriviallyNormedField 𝕜₁
inst✝⁹ : NormedSpace 𝕜₁ E
inst✝⁸ : NontriviallyNormedField 𝕜₂
inst✝⁷ : NormedSpace 𝕜₂ F
inst✝⁶ : NontriviallyNormedField 𝕜₃
inst✝⁵ : NormedSpace 𝕜₃ G
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₂₁ : 𝕜₂ →+* 𝕜₁
inst✝⁴ : RingHomInvPair σ₁₂ σ₂₁
inst✝³ : RingHomInvPair σ₂₁ σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜₁ →+* 𝕜₃
inst✝² : RingHomIsometric σ₁₃
inst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝ : RingHomIsometric σ₂₃
f : F →SL[σ₂₃] G
e : E ≃ₛₗᵢ[σ₁₂] F
r : ℝ≥0
⊢ ‖f ∘SL ↑↑e‖₊ ≤ r ↔ ‖f‖₊ ≤ r

Tactic:
simp [opNNNorm_le_iff, ← e.forall_congr_right]

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

Goal state before:
pp: case mp
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := ⋯
h_O : Continuous O
f' : E →L[𝕜] F
⊢ f' ∈ range toContinuousLinearMap → f' ∈ O ⁻¹' {x | IsUnit x}

Tactic:
rintro ⟨e', rfl⟩

Goal state after:
case mp
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f ↦ ↑e.symm ∘SL f
h_O : Continuous O
e' : E ≃L[𝕜] F
⊢ ↑e' ∈ O ⁻¹' {x | IsUnit x}
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s₁ s₂ t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ.real s₂ ≤ μ.real s₁
ht : t₁ ⊆ t₂
htμ : μ.real t₂ ≤ μ.real t₁
h₁ : μ s₂ ≠ ∞
h₂ : μ t₂ ≠ ∞
⊢ μ s₂ ≤ μ s₁

Tactic:
exact (ENNReal.toReal_le_toReal h₁ (measure_ne_top_of_subset hs h₁)).1 hsμ

Goal state after:
Module: Mathlib.Order.Filter.IndicatorFunction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : One β
l : Filter α
f : α → β
hf : ∀ᶠ (x : α) in l, f x ≠ 1
s t : Set α
h : s.mulIndicator f =ᶠ[l] t.mulIndicator f
⊢ s =ᶠ[l] t

Tactic:
by
  have : ∀ {s : Set α}, Function.mulSupport (s.mulIndicator f) =ᶠ[l] s := fun {s} ↦ by
    rw [mulSupport_mulIndicator]
    exact (hf.mono fun x hx ↦ and_iff_left hx).set_eq
  exact this.symm.trans <| h.mulSupport.trans this

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:
apply h.smul_measure
  simp

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:
simp_rw [Pi.zero_apply, ← hasFiniteIntegral_iff_edist]

Goal state after:
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
⊢ HasFiniteIntegral f μ ∧ HasFiniteIntegral g μ
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
⊢ (AEStronglyMeasurable f Measure.count ∧ Summable fun x ↦ ‖f x‖) ↔ Summable fun x ↦ ‖f x‖

Tactic:
and_iff_right_iff_imp

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
⊢ (Summable fun x ↦ ‖f x‖) → AEStronglyMeasurable f Measure.count
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case refine_2
α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f g : α → E
h : Disjoint (support f) (support g)
hf : StronglyMeasurable f
hg : StronglyMeasurable g
hfg : Integrable (f + g) μ
⊢ Integrable g μ

Tactic:
· rw [← indicator_add_eq_right h]; exact hfg.indicator hg.measurableSet_support

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

Goal state before:
pp: case pos
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
ha : μ {a} ≠ ∞
ha' : ‖f a‖ₑ ≠ ∞
hab : a ≤ b
⊢ IntegrableOn f (Ioc a b) μ ∧ True ↔ IntegrableOn f (Ioc a b) μ

Tactic:
and_true

Goal state after:
case pos
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
ha : μ {a} ≠ ∞
ha' : ‖f a‖ₑ ≠ ∞
hab : a ≤ b
⊢ IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioc a b) μ
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case pos
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
hab : a ≤ b
⊢ IntegrableOn f (Ico a b) μ ∧ True ↔ IntegrableOn f (Ico a b) μ

Tactic:
and_true

Goal state after:
case pos
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
hab : a ≤ b
⊢ IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ico a b) μ
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case pos
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
ha : μ {a} ≠ ∞
ha' : ‖f a‖ₑ ≠ ∞
hab : a < b
⊢ IntegrableOn f (Ioo a b) μ ∧ True ↔ IntegrableOn f (Ioo a b) μ

Tactic:
and_true

Goal state after:
case pos
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
ha : μ {a} ≠ ∞
ha' : ‖f a‖ₑ ≠ ∞
hab : a < b
⊢ IntegrableOn f (Ioo a b) μ ↔ IntegrableOn f (Ioo a b) μ
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case pos
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
hab : a < b
⊢ IntegrableOn f (Ioo a b) μ ∧ True ↔ IntegrableOn f (Ioo a b) μ

Tactic:
and_true

Goal state after:
case pos
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
hab : a < b
⊢ IntegrableOn f (Ioo a b) μ ↔ IntegrableOn f (Ioo a b) μ
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
⊢ IntegrableOn f (Ioi b) μ ∧ True ↔ IntegrableOn f (Ioi b) μ

Tactic:
and_true

Goal state after:
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
⊢ IntegrableOn f (Ioi b) μ ↔ IntegrableOn f (Ioi b) μ
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
⊢ IntegrableOn f (Iio b) μ ∧ True ↔ IntegrableOn f (Iio b) μ

Tactic:
and_true

Goal state after:
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
⊢ IntegrableOn f (Iio b) μ ↔ IntegrableOn f (Iio b) μ
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case refine_1
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 : ContinuousMultilinearMap 𝕜 E 𝕜
z : G
⊢ ‖f.smulRight z‖₊ ≤ ‖f‖₊ * ‖z‖₊

Tactic:
refine opNNNorm_le_iff.2 fun m => (nnnorm_smul_le _ _).trans ?_

Goal state after:
case refine_1
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 : ContinuousMultilinearMap 𝕜 E 𝕜
z : G
m : (i : ι) → E i
⊢ ‖LinearMap.id (f.toMultilinearMap m)‖₊ * ‖z‖₊ ≤ ‖f‖₊ * ‖z‖₊ * ∏ i, ‖m i‖₊
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: case neg
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
f : ι → X → E
hE : Nonempty E
inst✝ : IsCompletelyMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
hl : l.NeBot
e : E := Classical.choice hE
conv : Set X := {x | ∃ c, Tendsto (fun x_1 ↦ f x_1 x) l (𝓝 c)}
mconv : MeasurableSet conv
hconv : StronglyMeasurable fun x ↦ l.limUnder fun x_1 ↦ f x_1 ↑x
x : X
hx : x ∉ conv
⊢ (l.limUnder fun x_1 ↦ f x_1 x) = Function.extend Subtype.val (fun x ↦ l.limUnder fun x_1 ↦ f x_1 ↑x) (fun x ↦ e) x

Tactic:
Function.extend_val_apply' hx,

Goal state after:
case neg
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
f : ι → X → E
hE : Nonempty E
inst✝ : IsCompletelyMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
hl : l.NeBot
e : E := Classical.choice hE
conv : Set X := {x | ∃ c, Tendsto (fun x_1 ↦ f x_1 x) l (𝓝 c)}
mconv : MeasurableSet conv
hconv : StronglyMeasurable fun x ↦ l.limUnder fun x_1 ↦ f x_1 ↑x
x : X
hx : x ∉ conv
⊢ (l.limUnder fun x_1 ↦ f x_1 x) = e
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝¹ : PseudoMetrizableSpace ε
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ℕ → Set X
su : s ⊆ ⋃ n, u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
⊢ AEStronglyMeasurable f (μ.restrict s)

Tactic:
have : s = ⋃ n, u n ∩ s := by rw [← iUnion_inter]; exact (inter_eq_right.mpr su).symm

Goal state after:
X : Type u_1
ε : Type u_3
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝¹ : PseudoMetrizableSpace ε
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ℕ → Set X
su : s ⊆ ⋃ n, u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
this : s = ⋃ n, u n ∩ s
⊢ AEStronglyMeasurable f (μ.restrict s)
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: case pos
X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsClosed[inst✝³] s
hf : LocallyIntegrableOn f s μ
x : X
h : x ∈ s
t : Set X
ht_nhds : t ∈ 𝓝[s] x
ht_int : IntegrableOn f t μ
⊢ IntegrableAtFilter f (𝓝 x) (μ.restrict s)

Tactic:
obtain ⟨u, hu_o, hu_x, hu_sub⟩ := mem_nhdsWithin.mp ht_nhds

Goal state after:
case pos
X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsClosed[inst✝³] s
hf : LocallyIntegrableOn f s μ
x : X
h : x ∈ s
t : Set X
ht_nhds : t ∈ 𝓝[s] x
ht_int : IntegrableOn f t μ
u : Set X
hu_o : IsOpen[inst✝³] u
hu_x : x ∈ u
hu_sub : u ∩ s ⊆ t
⊢ IntegrableAtFilter f (𝓝 x) (μ.restrict s)
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
E : Type u_6
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : NormedAddCommGroup E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
f : X → E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := ⋯
this✝ : BorelSpace E
hbelow : BddBelow (f '' s)
habove : BddAbove (f '' s)
this : Bornology.IsBounded (f '' s)
C : ℝ
hC : ∀ x ∈ f '' s, ‖x‖ ≤ C
A : MemLp (fun x ↦ C) ∞ (μ.restrict s)
⊢ ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x‖ ≤ ‖C‖

Tactic:
apply ae_of_all _ fun y hy ↦ ?_

Goal state after:
X : Type u_1
E : Type u_6
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : NormedAddCommGroup E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
f : X → E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
hbelow : BddBelow (f '' s)
habove : BddAbove (f '' s)
this : Bornology.IsBounded (f '' s)
C : ℝ
hC : ∀ x ∈ f '' s, ‖x‖ ≤ C
A : MemLp (fun x ↦ C) ∞ (μ.restrict s)
y : X
hy : y ∈ s
⊢ ‖f y‖ ≤ ‖C‖
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case neg
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 = ∞
y : E
x✝ : y ∈ f.range
hy0 : ¬y = 0
⊢ ‖y‖ₑ ^ p.toReal * μ (⇑f ⁻¹' {y}) < ∞

Tactic:
· refine ENNReal.mul_lt_top ?_ (hf y hy0)
    exact ENNReal.rpow_lt_top_of_nonneg ENNReal.toReal_nonneg ENNReal.coe_ne_top

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

Goal state before:

Tactic:
‖T s‖ ≤ C * μ.real s := hT.2 s hs hμs
    _ ≤ C * μ'.real s := by
      simp only [measureReal_def]
      gcongr
      exact hμ's.ne

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
f : α →ₛ F
μ : Measure α
s : Set α
hs : MeasurableSet s
⊢ integral μ (piecewise s hs f 0) = integral (μ.restrict s) f

Tactic:
refine (integral_eq_sum_of_subset ?_).trans
      ((sum_congr rfl fun y hy => ?_).trans (integral_eq_sum_filter _ _).symm)
  · intro y hy
    simp only [mem_filter, mem_range, coe_piecewise, coe_zero, piecewise_eq_indicator,
      mem_range_indicator] at *
    rcases hy with ⟨⟨rfl, -⟩ | ⟨x, -, rfl⟩, h₀⟩
    exacts [(h₀ rfl).elim, ⟨Set.mem_range_self _, h₀⟩]
  · dsimp
    rw [Set.piecewise_eq_indicator, indicator_preimage_of_notMem,
      measureReal_restrict_apply (f.measurableSet_preimage _)]
    exact fun h₀ => (mem_filter.1 hy).2 (Eq.symm h₀)

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

Goal state before:
pp: case hbc
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
C : ℝ
hT_norm : ∀ (s : Set α), MeasurableSet s → ‖T s‖ ≤ C * μ.real s
f : α →ₛ F
x : F
a✝ : x ∈ f.range
⊢ ‖T (⇑f ⁻¹' {x})‖ ≤ C * μ.real (⇑f ⁻¹' {x})

Tactic:
exact hT_norm _ <| SimpleFunc.measurableSet_fiber _ _

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
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
⊢ ∀ ⦃f : α → E⦄, MemLp f p μ → motive f

Tactic:
have : ∀ f : SimpleFunc α E, MemLp f p μ → motive f := by
    apply SimpleFunc.induction
    · intro c s hs h
      by_cases hc : c = 0
      · subst hc; convert! indicator 0 MeasurableSet.empty (by simp) using 1; ext; simp
      have hp_pos : p ≠ 0 := (lt_of_lt_of_le zero_lt_one _i.elim).ne'
      exact indicator c hs (SimpleFunc.measure_lt_top_of_memLp_indicator hp_pos hp_ne_top hc hs h)
    · intro f g hfg hf hg int_fg
      rw [SimpleFunc.coe_add,
        memLp_add_of_disjoint hfg f.stronglyMeasurable g.stronglyMeasurable] at int_fg
      exact add hfg int_fg.1 int_fg.2 (hf int_fg.1) (hg int_fg.2)
  have : ∀ f : Lp.simpleFunc E p μ, motive f := by
    intro f
    exact
      ae (Lp.simpleFunc.toSimpleFunc_eq_toFun f) (Lp.simpleFunc.memLp f)
        (this (Lp.simpleFunc.toSimpleFunc f) (Lp.simpleFunc.memLp f))
  have : ∀ f : Lp E p μ, motive f := fun f =>
    (Lp.simpleFunc.denseRange hp_ne_top).induction_on f closed this
  exact fun f hf => ae hf.coeFn_toLp (Lp.memLp _) (this (hf.toLp f))

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
b : ℝ≥0
h : ∫⁻ (a : α), ↑(f a) ∂μ ≤ ↑b
hf : Integrable (fun a ↦ ↑(f a)) μ
⊢ ∫ (a : α), ↑(f a) ∂μ ≤ ↑b

Tactic:
exact (lintegral_coe_le_coe_iff_integral_le hf).1 h

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hμ : NeZero μ
hf : Integrable (fun x ↦ Real.exp (f x)) μ
this : (Function.support fun x ↦ Real.exp (f x)) = univ
⊢ 0 < μ (Function.support fun x ↦ Real.exp (f x))

Tactic:
by simp [this, hμ.out]

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 : (α → E) → Prop
h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → P (s.indicator fun x ↦ c)
h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Integrable f μ → Integrable g μ → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑f}
h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Integrable f μ → P f → P g
⊢ ∀ ⦃f : α → E⦄, Integrable f μ → P f

Tactic:
by
  simp only [← memLp_one_iff_integrable] at *
  exact MemLp.induction one_ne_top (motive := P) h_ind h_add h_closed h_ae

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁹ : Ring R
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module R V
inst✝⁶ : TopologicalSpace P
inst✝⁵ : AddTorsor V P
inst✝⁴ : AddCommGroup W
inst✝³ : Module R W
inst✝² : TopologicalSpace Q
inst✝¹ : AddTorsor W Q
e : P ≃ᴬ[R] Q
s : AffineSubspace R P
inst✝ : Nonempty ↥s
⊢ Continuous ((↑e).affineSubspaceMap s).toFun

Tactic:
by simpa [Topology.IsEmbedding.subtypeVal.continuous_iff] using!
      (e.continuous.comp continuous_subtype_val).congr fun _ => rfl

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
c : ℝ
f : ↥(Lp E 1 μ)
⊢ (setToL1 ⋯) f = c • (setToL1 hT) f

Tactic:
apply setToL1_unique (hT.smul c) (A := c • setToL1 hT) _ f

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
c : ℝ
f : ↥(Lp E 1 μ)
⊢ ∀ (f : ↥(α →₁ₛ[μ] E)), (setToL1SCLM α E μ ⋯) f = (c • setToL1 hT) ↑f
Module: Mathlib.MeasureTheory.Integral.Bochner.SumMeasure

Goal state before:
pp: X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSingletonClass X
μ : Measure X
f : X → E
s : Set X
hs : s.Countable
hf : IntegrableOn f s μ
hi : Countable { x // x ∈ s }
⊢ ∫ (x : X) in s, f x ∂μ = ∑' (x : ↑s), μ.real {↑x} • f ↑x

Tactic:
have hf' : Integrable (fun (x : s) ↦ f x) (Measure.comap Subtype.val μ) := by
    rw [IntegrableOn, ← map_comap_subtype_coe, integrable_map_measure] at hf
    · apply hf
    · exact Integrable.aestronglyMeasurable hf
    · exact Measurable.aemeasurable measurable_subtype_coe
    · exact Countable.measurableSet hs

Goal state after:
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSingletonClass X
μ : Measure X
f : X → E
s : Set X
hs : s.Countable
hf : IntegrableOn f s μ
hi : Countable { x // x ∈ s }
hf' : Integrable (fun x ↦ f ↑x) (Measure.comap Subtype.val μ)
⊢ ∫ (x : X) in s, f x ∂μ = ∑' (x : ↑s), μ.real {↑x} • f ↑x