mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureProd

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
β : Type u_2
inst✝² : MeasurableSpace β
μ : FiniteMeasure α
ν : FiniteMeasure β
α' : Type u_3
inst✝¹ : MeasurableSpace α'
β' : Type u_4
inst✝ : MeasurableSpace β'
f : α → α'
g : β → β'
f_mble : Measurable f
g_mble : Measurable g
⊢ (μ.map f).prod (ν.map g) = (μ.prod ν).map (Prod.map f g)

Tactic:
by
  apply Subtype.ext
  simp only [val_eq_toMeasure, toMeasure_prod, toMeasure_map]
  rw [Measure.map_prod_map _ _ f_mble g_mble]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.DistribChar

Goal state before:
pp: case e_a
G : Type u_1
A : Type u_2
inst✝⁶ : Group G
inst✝⁵ : AddCommGroup A
inst✝⁴ : DistribMulAction G A
inst✝³ : TopologicalSpace A
inst✝² : IsTopologicalAddGroup A
inst✝¹ : LocallyCompactSpace A
inst✝ : ContinuousConstSMul G A
g✝ : G
this✝ : MeasurableSpace A := borel A
this : BorelSpace A
g g' : G
⊢ ((DomMulAct.mk g' * DomMulAct.mk g) • addHaar).addHaarScalarFactor (DomMulAct.mk g' • addHaar) =
    (DomMulAct.mk g • addHaar).addHaarScalarFactor addHaar

Tactic:
simp_rw [mul_smul]

Goal state after:
case e_a
G : Type u_1
A : Type u_2
inst✝⁶ : Group G
inst✝⁵ : AddCommGroup A
inst✝⁴ : DistribMulAction G A
inst✝³ : TopologicalSpace A
inst✝² : IsTopologicalAddGroup A
inst✝¹ : LocallyCompactSpace A
inst✝ : ContinuousConstSMul G A
g✝ : G
this✝ : MeasurableSpace A := borel A
this : BorelSpace A
g g' : G
⊢ (DomMulAct.mk g' • DomMulAct.mk g • addHaar).addHaarScalarFactor (DomMulAct.mk g' • addHaar) =
    (DomMulAct.mk g • addHaar).addHaarScalarFactor addHaar
Module: Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar

Goal state before:
pp: G : Type u_1
inst✝⁶ : Group G
inst✝⁵ : TopologicalSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : IsTopologicalGroup G
inst✝¹ : LocallyCompactSpace G
inst✝ : CompactSpace G
φ : G ≃ₜ* G
μ : Measure G := haarMeasure { carrier := univ, isCompact' := ⋯, interior_nonempty' := ⋯ }
⊢ mulEquivHaarChar φ = 1

Tactic:
have hμ : μ univ = 1 := haarMeasure_self

Goal state after:
G : Type u_1
inst✝⁶ : Group G
inst✝⁵ : TopologicalSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : IsTopologicalGroup G
inst✝¹ : LocallyCompactSpace G
inst✝ : CompactSpace G
φ : G ≃ₜ* G
μ : Measure G := haarMeasure { carrier := univ, isCompact' := ⋯, interior_nonempty' := ⋯ }
hμ : μ univ = 1
⊢ mulEquivHaarChar φ = 1
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
g : E →C_c ℝ
⊢ ∃ x ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖),
    Tendsto (fun μ ↦ ∫ (x : E), g x ∂↑μ) (↑f) (𝓝 x)

Tactic:
simp only [Tendsto, ← Ultrafilter.coe_map]

Goal state after:
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
g : E →C_c ℝ
⊢ ∃ x ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖),
    ↑(Ultrafilter.map (fun μ ↦ ∫ (x : E), g x ∂↑μ) f) ≤ 𝓝 x
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: case pos
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
h : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
y : E
this : ProperSpace E
hy : y = 0
⊢ Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖0‖}) atTop (𝓝 0)

Tactic:
refine (tendsto_congr' ?_).mpr tendsto_const_nhds

Goal state after:
case pos
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
h : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
y : E
this : ProperSpace E
hy : y = 0
⊢ (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖0‖}) =ᶠ[atTop] fun x ↦ 0
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: case pos
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : y = 0
⊢ Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖0‖}) atTop) atTop (𝓝 0)

Tactic:
refine (tendsto_congr' ?_).mpr tendsto_const_nhds

Goal state after:
case pos
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : y = 0
⊢ (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖0‖}) atTop) =ᶠ[atTop] fun x ↦ 0
Module: Mathlib.MeasureTheory.Measure.MeasuredSets

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
⊢ ∀ t ∈ C, ∀ (ε : ℝ≥0∞), 0 < ε → ∃ t_1 ∈ C, μ (t_1 ∆ t) < ε

Tactic:
intro s hs ε εpos

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s✝ : Set α
hs✝ : MeasurableSet s✝
ε✝ : ℝ≥0∞
hε : 0 < ε✝
s : Set α
hs : s ∈ C
ε : ℝ≥0∞
εpos : 0 < ε
⊢ ∃ t ∈ C, μ (t ∆ s) < ε
Module: Mathlib.MeasureTheory.Measure.Support

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : MeasurableSpace X
μ : Measure X
⊢ μ.supportᶜ = ⋃₀ {t | IsOpen[inst✝¹] t ∧ μ t = 0}

Tactic:
ext x
  simp only [Set.mem_compl_iff, Set.mem_sUnion, Set.mem_ofPred_eq, and_right_comm,
    nhds_basis_opens x |>.notMem_measureSupport, fun t ↦ and_comm (b := x ∈ t)]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
n : ℕ
h : Monotone K
this : Tendsto (fun m ↦ ∑ i ∈ Finset.range (m + 1), ↑(ν i) (K n)ᶜ) atTop (𝓝 (∑' (i : ℕ), ↑(ν i) (K n)ᶜ))
m : ℕ
hm : n ≤ m
⊢ ∀ x ≤ m, x ≤ n → (ν x) (K n)ᶜ = 0

Tactic:
intro i hi h'i

Goal state after:
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
n : ℕ
h : Monotone K
this : Tendsto (fun m ↦ ∑ i ∈ Finset.range (m + 1), ↑(ν i) (K n)ᶜ) atTop (𝓝 (∑' (i : ℕ), ↑(ν i) (K n)ᶜ))
m : ℕ
hm : n ≤ m
i : ℕ
hi : i ≤ m
h'i : i ≤ n
⊢ (ν i) (K n)ᶜ = 0
Module: Mathlib.MeasureTheory.Measure.SubFinite

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
inst✝ : IsFiniteMeasure (μ.withDensity g)
hg : Measurable g
hgf : g ≤ᵐ[μ] f
s : Set α
hs : MeasurableSet s
⊢ (μ.withDensity (f - g)) s = (μ.withDensity f - μ.withDensity g) s

Tactic:
rw [sub_apply hs (withDensity_mono hgf), withDensity_apply _ hs, withDensity_apply _ hs,
    withDensity_apply _ hs, ← lintegral_sub hg _ (ae_restrict_of_ae hgf)]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
inst✝ : IsFiniteMeasure (μ.withDensity g)
hg : Measurable g
hgf : g ≤ᵐ[μ] f
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, (f - g) a ∂μ = ∫⁻ (a : α) in s, f a - g a ∂μ

α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
inst✝ : IsFiniteMeasure (μ.withDensity g)
hg : Measurable g
hgf : g ≤ᵐ[μ] f
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, g a ∂μ ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: case mp.inr
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
s : Set α
h : μ s = 1
h₁ : μ univ = 1
⊢ μ univ = 1

Tactic:
· exact h₁

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: case refine_2
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, m s = ∞
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ i, f i
A : ℕ → Finset (Set α) := fun i ↦ hC.disjointOfDiff ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ ↑(A i)
h_m_eq : ∀ (i : ℕ), m (f i) = m (f i ∩ s) + ∑ u ∈ A i, m u
⊢ (OuterMeasure.ofFunction ⇑m ⋯) (t \ s) ≤ ∑' (a : ℕ), ∑ u ∈ A a, m u

Tactic:
· apply le_trans <| (OuterMeasure.ofFunction m addContent_empty).mono
      <| (iUnion_sdiff s f) ▸ sdiff_subset_sdiff_left hf_subset
    simp only [OuterMeasure.measureOf_eq_coe, A]
    apply le_trans <| measure_iUnion_le (μ := OuterMeasure.ofFunction m addContent_empty)
      (fun i ↦ f i \ s)
    apply ENNReal.tsum_le_tsum
    intro i
    simp_rw [sUnion_eq_biUnion] at h_diff_eq_sUnion
    rw [h_diff_eq_sUnion]
    obtain h6 := MeasureTheory.measure_biUnion_finset_le
      (μ := OuterMeasure.ofFunction m addContent_empty) (A i) id
    simp only [id_eq] at h6
    exact le_trans h6 <| Finset.sum_le_sum <| fun b _ ↦ OuterMeasure.ofFunction_le b

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: case iUnion
α : Type u_1
C : Set (Set α)
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
s : Set α
f : ℕ → Set α
a✝ : ∀ (n : ℕ), MeasurableSpace.GenerateMeasurable C (f n)
h : ∀ (n : ℕ), (inducedOuterMeasure (fun x x_1 ↦ m x) ⋯ ⋯).IsCaratheodory (f n)
⊢ (inducedOuterMeasure (fun x x_1 ↦ m x) ⋯ ⋯).IsCaratheodory (⋃ i, f i)

Tactic:
exact OuterMeasure.isCaratheodory_iUnion _ h

Goal state after:
Module: Mathlib.MeasureTheory.SpecificCodomains.WithLp

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hf : Integrable f μ
⊢ (∫ (x : X), (WithLp.prodContinuousLinearEquiv q ℝ E F) (f x) ∂μ).1 = ∫ (x : X), (f x).fst ∂μ

Tactic:
fst_integral

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hf : Integrable f μ
⊢ ∫ (x : X), ((WithLp.prodContinuousLinearEquiv q ℝ E F) (f x)).1 ∂μ = ∫ (x : X), (f x).fst ∂μ

X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hf : Integrable f μ
⊢ Integrable (fun x ↦ (WithLp.prodContinuousLinearEquiv q ℝ E F) (f x)) μ
Module: Mathlib.MeasureTheory.SpecificCodomains.WithLp

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
hf : Integrable f μ
⊢ (∫ (x : X), f x ∂μ).snd = ∫ (x : X), (f x).snd ∂μ

Tactic:
by
  rw [← WithLp.ofLp_snd]
  conv => enter [1, 1]; change WithLp.prodContinuousLinearEquiv q ℝ E F _
  rw [← ContinuousLinearEquiv.integral_comp_comm, snd_integral]
  · rfl
  · exact (ContinuousLinearEquiv.integrable_comp_iff _).2 hf

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: case refine_3
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
a : ℝ≥0∞
ha : a < preVariationFun (fun x ↦ ‖μ x‖ₑ) s
P : Finpartition ⟨s, hs⟩
hP : a < ∑ p ∈ P.parts, (fun x ↦ ‖μ x‖ₑ) ↑p
⊢ ∀ t ∈ Finset.map (Function.Embedding.subtype MeasurableSet) P.parts, MeasurableSet t

pp: case refine_4
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
a : ℝ≥0∞
ha : a < preVariationFun (fun x ↦ ‖μ x‖ₑ) s
P : Finpartition ⟨s, hs⟩
hP : a < ∑ p ∈ P.parts, (fun x ↦ ‖μ x‖ₑ) ↑p
⊢ a < ∑ p ∈ Finset.map (Function.Embedding.subtype MeasurableSet) P.parts, ‖μ p‖ₑ

Tactic:
· simp +contextual

Goal state after:
case refine_4
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
a : ℝ≥0∞
ha : a < preVariationFun (fun x ↦ ‖μ x‖ₑ) s
P : Finpartition ⟨s, hs⟩
hP : a < ∑ p ∈ P.parts, (fun x ↦ ‖μ x‖ₑ) ↑p
⊢ a < ∑ p ∈ Finset.map (Function.Embedding.subtype MeasurableSet) P.parts, ‖μ p‖ₑ
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: case refine_3
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
hμ : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≠ ∞
P : Finpartition ⟨s, hs⟩
hP : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≤ ∑ p ∈ P.parts, (fun x ↦ ‖μ x‖ₑ) ↑p + ε
⊢ ∀ t ∈ Finset.map (Function.Embedding.subtype MeasurableSet) P.parts, MeasurableSet t

pp: case refine_4
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
hμ : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≠ ∞
P : Finpartition ⟨s, hs⟩
hP : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≤ ∑ p ∈ P.parts, (fun x ↦ ‖μ x‖ₑ) ↑p + ε
⊢ preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≤ ∑ x ∈ Finset.map (Function.Embedding.subtype MeasurableSet) P.parts, ‖μ x‖ₑ + ε

Tactic:
· simp +contextual

Goal state after:
case refine_4
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
hμ : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≠ ∞
P : Finpartition ⟨s, hs⟩
hP : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≤ ∑ p ∈ P.parts, (fun x ↦ ‖μ x‖ₑ) ↑p + ε
⊢ preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≤ ∑ x ∈ Finset.map (Function.Embedding.subtype MeasurableSet) P.parts, ‖μ x‖ₑ + ε
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
t : Set X
ht : MeasurableSet t
⊢ μ.variation (t ∩ s) ≤ (μ.restrict s).variation t

Tactic:
calc μ.variation (t ∩ s)
    _ ≤ (μ.restrict s).variation (t ∩ s) := by
      apply variation_apply_le_of_forall_enorm_le (ht.inter hs) (fun u u_meas hu ↦ ?_)
      have : μ u = μ.restrict s u :=
        (VectorMeasure.restrict_eq_self _ hs u_meas (hu.trans inter_subset_right)).symm
      rw [this]
      apply enorm_measure_le_variation
    _ ≤ (μ.restrict s).variation t := by
      gcongr
      exact Set.inter_subset_left

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
hs : MeasurableSet s
hfi : μ.IntegrableOn f s
⊢ s.indicator f =ᵐ[μ.variation.restrict s] f

Tactic:
exact indicator_ae_eq_restrict hs

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case pos
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
μ : VectorMeasure X F
s : Set X
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace ℝ F
inst✝² : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : CompleteSpace G
inst✝ : IsFiniteMeasure (μ.variation.restrict s)
c : E
hs : MeasurableSet s
⊢ ∫ᵛ (x : X) in s, c ∂[B; μ] = (B c) (μ s)

Tactic:
have : IsFiniteMeasure (μ.restrict s).variation := by
      rwa [variation_restrict hs]
    rw [integral_const, restrict_apply _ hs MeasurableSet.univ, univ_inter]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Semivariation

Goal state before:
pp: X : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
mX : MeasurableSpace X
μ : VectorMeasure X E
h : μ.semivariation univ = ∞
t : Set X → Set X
t_meas : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → MeasurableSet (t s)
t_subs : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → t s ⊆ s
t_var : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → μ.semivariation (t s) = ∞
ht : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → 1 ≤ ‖μ (s \ t s)‖ₑ
s : ℕ → Set X := fun n ↦ t^[n] univ
hs : ∀ (n : ℕ), MeasurableSet (s n) ∧ μ.semivariation (s n) = ∞
u : ℕ → Set X := fun n ↦ s n \ s (n + 1)
n : ℕ
⊢ 1 ≤ ‖μ (t^[n] univ \ t (t^[n] univ))‖ₑ

Tactic:
exact ht _ (hs n).1 (hs n).2

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.AddContent

Goal state before:
pp: α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : hα = generateFrom C
M : ∀ s ∈ C, MeasurableSet s
D : Set (Set α)
DC : D ⊆ C
D_count : D.Countable
hD : ∀ s ∈ C, s ≤ᵐ[μ] ⋃₀ D
MD : MeasurableSet (⋃₀ D)
μ' : Measure α := μ.restrict (⋃₀ D)
⊢ ∃ m', (∀ s ∈ C, m' s = m s) ∧ ∀ (s : Set α), ‖m' s‖ₑ ≤ μ' s

Tactic:
apply exists_extension_of_isSetSemiring_of_le_measure_of_generateFrom_of_cover hC
      (fun s hs ↦ ?_) h'C ?_

Goal state after:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : hα = generateFrom C
M : ∀ s ∈ C, MeasurableSet s
D : Set (Set α)
DC : D ⊆ C
D_count : D.Countable
hD : ∀ s ∈ C, s ≤ᵐ[μ] ⋃₀ D
MD : MeasurableSet (⋃₀ D)
μ' : Measure α := μ.restrict (⋃₀ D)
⊢ ∃ D, D.Countable ∧ D ⊆ C ∧ μ' (⋃₀ D)ᶜ = 0

α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : hα = generateFrom C
M : ∀ s ∈ C, MeasurableSet s
D : Set (Set α)
DC : D ⊆ C
D_count : D.Countable
hD : ∀ s ∈ C, s ≤ᵐ[μ] ⋃₀ D
MD : MeasurableSet (⋃₀ D)
μ' : Measure α := μ.restrict (⋃₀ D)
s : Set α
hs : s ∈ C
⊢ ‖m s‖ₑ ≤ μ' s
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case pos.h_closed
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup H
μ : VectorMeasure X F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
f : X → H
C : H →L[ℝ] E
hf : Integrable f μ.variation
hG : CompleteSpace G
⊢ IsClosed {f | ∫ᵛ (y : X), C (↑↑f y) ∂[B; μ] = ∫ᵛ (y : X), ↑↑f y ∂[B ∘SL C; μ]}

pp: case pos.h_ae
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup H
μ : VectorMeasure X F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
f : X → H
C : H →L[ℝ] E
hf : Integrable f μ.variation
hG : CompleteSpace G
⊢ ∀ ⦃f g : X → H⦄,
    f =ᵐ[μ.variation] g →
      Integrable f μ.variation →
        ∫ᵛ (y : X), C (f y) ∂[B; μ] = ∫ᵛ (y : X), f y ∂[B ∘SL C; μ] →
          ∫ᵛ (y : X), C (g y) ∂[B; μ] = ∫ᵛ (y : X), g y ∂[B ∘SL C; μ]

Tactic:
· apply isClosed_eq
    · have I (f : Lp H 1 μ.variation) : ∫ᵛ x, C (f x) ∂[B; μ] = ∫ᵛ x, (C.compLp f) x ∂[B; μ] :=
        (integral_congr_ae (coeFn_compLp _ _)).symm
      simp_rw [I]
      exact continuous_integral.comp (C.compLpL 1 μ.variation).continuous
    · exact continuous_integral

Goal state after:
case pos.h_ae
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup H
μ : VectorMeasure X F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
f : X → H
C : H →L[ℝ] E
hf : Integrable f μ.variation
hG : CompleteSpace G
⊢ ∀ ⦃f g : X → H⦄,
    f =ᵐ[μ.variation] g →
      Integrable f μ.variation →
        ∫ᵛ (y : X), C (f y) ∂[B; μ] = ∫ᵛ (y : X), f y ∂[B ∘SL C; μ] →
          ∫ᵛ (y : X), C (g y) ∂[B; μ] = ∫ᵛ (y : X), g y ∂[B ∘SL C; μ]