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

Goal state before:
pp: case inr.refine_2
α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
hf : BoundedVariationOn f univ
h'α : Nonempty α
m : AddContent E {s | ∃ u v, u ≤ v ∧ s = Ioc u v} := AddContent.onIoc (Function.rightLim f)
A : ∀ s ∈ {s | ∃ u v, u ≤ v ∧ s = Ioc u v}, ‖m s‖ₑ ≤ hf.measureAux s
B : hα = generateFrom {s | ∃ u v, u ≤ v ∧ s = Ioc u v}
m' : VectorMeasure α E
hm' : ∀ s ∈ {s | ∃ u v, u ≤ v ∧ s = Ioc u v}, m' s = m s
h'm' : ∀ (s : Set α), ‖m' s‖ₑ ≤ hf.measureAux s
⊢ m' botSet = 0

Tactic:
apply enorm_eq_zero.1

Goal state after:
case inr.refine_2
α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
hf : BoundedVariationOn f univ
h'α : Nonempty α
m : AddContent E {s | ∃ u v, u ≤ v ∧ s = Ioc u v} := ⋯
A : ∀ s ∈ {s | ∃ u v, u ≤ v ∧ s = Ioc u v}, ‖m s‖ₑ ≤ hf.measureAux s
B : hα = generateFrom {s | ∃ u v, u ≤ v ∧ s = Ioc u v}
m' : VectorMeasure α E
hm' : ∀ s ∈ {s | ∃ u v, u ≤ v ∧ s = Ioc u v}, m' s = m s
h'm' : ∀ (s : Set α), ‖m' s‖ₑ ≤ hf.measureAux s
⊢ ‖m' botSet‖ₑ = 0
Module: Mathlib.MeasureTheory.VectorMeasure.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
a : α
hf : BoundedVariationOn f univ
ha : ¬IsBot a
b : α
hb : b < a
u : ℕ → α
u_mono : StrictMono u
u_lim : Tendsto u atTop (𝓝 a)
u_lt_a : ∀ (n : ℕ), u n < a
A : {a} = ⋂ n, Ioc (u n) a
L1 : Tendsto (fun n ↦ hf.vectorMeasure (Ioc (u n) a)) atTop (𝓝 (hf.vectorMeasure {a}))
⊢ Tendsto (fun n ↦ hf.vectorMeasure (Ioc (u n) a)) atTop (𝓝 (Function.rightLim f a - Function.leftLim f a))

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
a : α
hf : BoundedVariationOn f univ
ha : ¬IsBot a
b : α
hb : b < a
u : ℕ → α
u_mono : StrictMono u
u_lim : Tendsto u atTop (𝓝 a)
u_lt_a : ∀ (n : ℕ), u n < a
A : {a} = ⋂ n, Ioc (u n) a
L1 : Tendsto (fun n ↦ hf.vectorMeasure (Ioc (u n) a)) atTop (𝓝 (hf.vectorMeasure {a}))
⊢ Tendsto (fun n ↦ hf.vectorMeasure (Ioc (u n) a)) atTop (𝓝 (Function.rightLim f a - Function.leftLim f a))
Module: Mathlib.MeasureTheory.VectorMeasure.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
hf : BoundedVariationOn f univ
a : α
this✝ : Nonempty α
hlim : Tendsto f atTop (𝓝 (atTop.limUnder f))
u : ℕ → α
u_mono : Monotone u
hu : Tendsto u atTop atTop
this : Ici a = ⋃ n, Icc a (u n)
⊢ Tendsto (fun n ↦ hf.vectorMeasure (Icc a (u n))) atTop (𝓝 (hf.vectorMeasure (Ici a)))

Tactic:
this

Goal state after:
α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
hf : BoundedVariationOn f univ
a : α
this✝ : Nonempty α
hlim : Tendsto f atTop (𝓝 (atTop.limUnder f))
u : ℕ → α
u_mono : Monotone u
hu : Tendsto u atTop atTop
this : Ici a = ⋃ n, Icc a (u n)
⊢ Tendsto (fun n ↦ hf.vectorMeasure (Icc a (u n))) atTop (𝓝 (hf.vectorMeasure (⋃ n, Icc a (u n))))
Module: Mathlib.MeasureTheory.VectorMeasure.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
hf : BoundedVariationOn f univ
a : α
this✝ : Nonempty α
hlim : Tendsto f atBot (𝓝 (atBot.limUnder f))
u : ℕ → α
u_anti : Antitone u
hu : Tendsto u atTop atBot
this : Iic a = ⋃ n, Icc (u n) a
⊢ Tendsto (fun n ↦ hf.vectorMeasure (Icc (u n) a)) atTop (𝓝 (hf.vectorMeasure (Iic a)))

Tactic:
this

Goal state after:
α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
hf : BoundedVariationOn f univ
a : α
this✝ : Nonempty α
hlim : Tendsto f atBot (𝓝 (atBot.limUnder f))
u : ℕ → α
u_anti : Antitone u
hu : Tendsto u atTop atBot
this : Iic a = ⋃ n, Icc (u n) a
⊢ Tendsto (fun n ↦ hf.vectorMeasure (Icc (u n) a)) atTop (𝓝 (hf.vectorMeasure (⋃ n, Icc (u n) a)))
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 : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, μ (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s | ∃ c ∈ C, ↑s = c}
C'C : ∀ s ∈ C', ↑s ∈ C
C'_dense : Dense C'
m₀ : ↑C' → E := fun x ↦ m ↑↑x
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ μ ↑s
s t : MeasuredSets μ
h : Disjoint ↑s ↑t
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
hδ : 8 * δ = ε
s' : Set α
s'C : s' ∈ C
hs' : μ (s' ∆ ↑s) < δ
t' : Set α
t'C : t' ∈ C
ht' : μ (t' ∆ ↑t) < δ
It : ‖m t' - m₁ t‖ₑ < δ
hμ' : μ (s' ∩ t') < 2 * δ
s'' : Set α := s' \ t'
s''C : s'' ∈ C
hs'' : μ (s'' ∆ ↑s) < 3 * δ
Is : ‖m s'' - m₁ s‖ₑ < 3 * δ
Ist : ‖m (s'' ∪ t') - m₁ ⟨↑s ∪ ↑t, ⋯⟩‖ₑ < 4 * δ
⊢ ‖m (s'' ∪ t') - m s'' - m t'‖ₑ + ‖m₁ ⟨↑s ∪ ↑t, ⋯⟩ - m (s'' ∪ t')‖ₑ + ‖m s'' - m₁ s‖ₑ + ‖m t' - m₁ t‖ₑ =
    ‖m₁ ⟨↑s ∪ ↑t, ⋯⟩ - m (s'' ∪ t')‖ₑ + ‖m s'' - m₁ s‖ₑ + ‖m t' - m₁ t‖ₑ

Tactic:
rw [addContent_union hC s''C t'C Set.disjoint_sdiff_left]

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 : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, μ (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s | ∃ c ∈ C, ↑s = c}
C'C : ∀ s ∈ C', ↑s ∈ C
C'_dense : Dense C'
m₀ : ↑C' → E := fun x ↦ m ↑↑x
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ μ ↑s
s t : MeasuredSets μ
h : Disjoint ↑s ↑t
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
hδ : 8 * δ = ε
s' : Set α
s'C : s' ∈ C
hs' : μ (s' ∆ ↑s) < δ
t' : Set α
t'C : t' ∈ C
ht' : μ (t' ∆ ↑t) < δ
It : ‖m t' - m₁ t‖ₑ < δ
hμ' : μ (s' ∩ t') < 2 * δ
s'' : Set α := s' \ t'
s''C : s'' ∈ C
hs'' : μ (s'' ∆ ↑s) < 3 * δ
Is : ‖m s'' - m₁ s‖ₑ < 3 * δ
Ist : ‖m (s'' ∪ t') - m₁ ⟨↑s ∪ ↑t, ⋯⟩‖ₑ < 4 * δ
⊢ ‖m s'' + m t' - m s'' - m t'‖ₑ + ‖m₁ ⟨↑s ∪ ↑t, ⋯⟩ - (m s'' + m t')‖ₑ + ‖m s'' - m₁ s‖ₑ + ‖m t' - m₁ t‖ₑ =
    ‖m₁ ⟨↑s ∪ ↑t, ⋯⟩ - (m s'' + m t')‖ₑ + ‖m s'' - m₁ s‖ₑ + ‖m t' - m₁ t‖ₑ
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.JordanSub

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ ν : Measure X
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
⊢ (μ.jordanDecompositionOfToSignedMeasureSub ν).toSignedMeasure = μ.toSignedMeasure - ν.toSignedMeasure

Tactic:
[

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ ν : Measure X
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
⊢ (μ.jordanDecompositionOfToSignedMeasureSub ν).toSignedMeasure = μ.toSignedMeasure - ν.toSignedMeasure
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
h : ∀ᵐ (x : X) ∂μ.variation, x ∉ s → f x = 0
⊢ ∫ᵛ (x : X) in s, f x ∂[B; μ] = ∫ᵛ (x : X), f x ∂[B; μ]

Tactic:
by
  symm
  nth_rw 1 [← setIntegral_univ]
  apply setIntegral_eq_of_subset_of_ae_sdiff_eq_zero hs MeasurableSet.univ (subset_univ _)
  apply (ae_restrict_iff' (MeasurableSet.univ.diff hs)).2
  filter_upwards [h] with x hx h'x using hx h'x.2

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

Goal state before:
pp: case compl
X : Type u_2
Y : Type u_3
F : Type u_5
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝ : NormedAddCommGroup F
ν : VectorMeasure Y F
s✝ s : Set (X × Y)
hs : MeasurableSet s
ihs : StronglyMeasurable fun x ↦ ν (Prod.mk x ⁻¹' s)
⊢ StronglyMeasurable fun x ↦ ν (Prod.mk x ⁻¹' sᶜ)

Tactic:
[

Goal state after:
case compl
X : Type u_2
Y : Type u_3
F : Type u_5
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝ : NormedAddCommGroup F
ν : VectorMeasure Y F
s✝ s : Set (X × Y)
hs : MeasurableSet s
ihs : StronglyMeasurable fun x ↦ ν (Prod.mk x ⁻¹' s)
⊢ StronglyMeasurable fun x ↦ ν (Prod.mk x ⁻¹' sᶜ)
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case pos.h_ind
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
c : H
s : Set X
hs : MeasurableSet s
hc : μ.variation s < ∞
this✝ : IsFiniteMeasure (μ.variation.restrict s)
this : (fun y ↦ C (s.indicator (fun x ↦ c) y)) = s.indicator fun x ↦ C c
⊢ ∫ᵛ (y : X), C (s.indicator (fun x ↦ c) y) ∂[B; μ] = ((B ∘SL C) c) (μ s)

Tactic:
[

Goal state after:
case pos.h_ind
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
c : H
s : Set X
hs : MeasurableSet s
hc : μ.variation s < ∞
this✝ : IsFiniteMeasure (μ.variation.restrict s)
this : (fun y ↦ C (s.indicator (fun x ↦ c) y)) = s.indicator fun x ↦ C c
⊢ ∫ᵛ (y : X), C (s.indicator (fun x ↦ c) y) ∂[B; μ] = ((B ∘SL C) c) (μ s)
Module: Mathlib.MeasureTheory.VectorMeasure.Prod

Goal state before:
pp: ι : Type u_1
X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
H : Type u_7
I : Type u_8
J : Type u_9
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : NormedAddCommGroup H
inst✝⁵ : NormedSpace ℝ H
inst✝⁴ : NormedAddCommGroup I
inst✝³ : NormedSpace ℝ I
inst✝² : NormedAddCommGroup J
inst✝¹ : NormedSpace ℝ J
μ✝ : VectorMeasure X E
ν✝ : VectorMeasure Y F
B✝ : E →L[ℝ] F →L[ℝ] G
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : IsFiniteMeasure μ.variation
f : ℕ → Set (X × Y)
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Disjoint on f)
A :
  ∀ (a : Finset ℕ),
    ∑ y ∈ a, ∫ᵛ (x : X), ν (Prod.mk x ⁻¹' f y) ∂[B.flip; μ] = ∫ᵛ (x : X), ∑ y ∈ a, ν (Prod.mk x ⁻¹' f y) ∂[B.flip; μ]
⊢ Tendsto (fun s ↦ ∑ x ∈ s, ∫ᵛ (x_1 : X), ν (Prod.mk x_1 ⁻¹' f x) ∂[B.flip; μ]) atTop
    (𝓝 ∫ᵛ (x : X), ν (⋃ i, Prod.mk x ⁻¹' f i) ∂[B.flip; μ])

Tactic:
[

Goal state after:
ι : Type u_1
X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
H : Type u_7
I : Type u_8
J : Type u_9
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : NormedAddCommGroup H
inst✝⁵ : NormedSpace ℝ H
inst✝⁴ : NormedAddCommGroup I
inst✝³ : NormedSpace ℝ I
inst✝² : NormedAddCommGroup J
inst✝¹ : NormedSpace ℝ J
μ✝ : VectorMeasure X E
ν✝ : VectorMeasure Y F
B✝ : E →L[ℝ] F →L[ℝ] G
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : IsFiniteMeasure μ.variation
f : ℕ → Set (X × Y)
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Disjoint on f)
A :
  ∀ (a : Finset ℕ),
    ∑ y ∈ a, ∫ᵛ (x : X), ν (Prod.mk x ⁻¹' f y) ∂[B.flip; μ] = ∫ᵛ (x : X), ∑ y ∈ a, ν (Prod.mk x ⁻¹' f y) ∂[B.flip; μ]
⊢ Tendsto (fun s ↦ ∑ x ∈ s, ∫ᵛ (x_1 : X), ν (Prod.mk x_1 ⁻¹' f x) ∂[B.flip; μ]) atTop
    (𝓝 ∫ᵛ (x : X), ν (⋃ i, Prod.mk x ⁻¹' f i) ∂[B.flip; μ])
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case pos.h_ind
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
c : H
s : Set X
hs : MeasurableSet s
hc : μ.variation s < ∞
this✝ : IsFiniteMeasure (μ.variation.restrict s)
this : (fun y ↦ C (s.indicator (fun x ↦ c) y)) = s.indicator fun x ↦ C c
⊢ ∫ᵛ (y : X), C (s.indicator (fun x ↦ c) y) ∂[B; μ] = ((B ∘SL C) c) (μ s)

Tactic:
this

Goal state after:
case pos.h_ind
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
c : H
s : Set X
hs : MeasurableSet s
hc : μ.variation s < ∞
this✝ : IsFiniteMeasure (μ.variation.restrict s)
this : (fun y ↦ C (s.indicator (fun x ↦ c) y)) = s.indicator fun x ↦ C c
⊢ μ.integral (s.indicator fun x ↦ C c) B = ((B ∘SL C) c) (μ s)
Module: Mathlib.MeasureTheory.VectorMeasure.AddContent

Goal state before:
pp: α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : CompleteSpace E
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ s ∈ C, MeasurableSet s
M : MeasurableSpace α := generateFrom C
Mle : M ≤ hα
μ' : Measure α := μ.trim Mle
hμ' : μ' = μ.trim Mle
m' : VectorMeasure α E
m'C : ∀ s ∈ C, m' s = m s
hm' : ∀ (s : Set α), ‖m' s‖ₑ ≤ μ' s
m'_le : m'.variation ≤ μ'
this✝ : MeasurableSpace α := hα
f : ℕ → Set α
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf : Pairwise (Function.onFun Disjoint f)
this :
  ∫ᵛ (x : α), μ[fun x ↦ ∑' (d : ℕ), (f d).indicator 1 x | M] x ∂•m' =
    ∫ᵛ (x : α), ∑' (d : ℕ), μ[(f d).indicator 1 | M] x ∂•m'
A : ∀ (i : ℕ), ∫⁻ (a : α), ‖μ[(f i).indicator 1 | M] a‖ₑ ∂μ = μ (f i)
⊢ ∑' (i : ℕ), ∫⁻ (a : α), ‖μ[(f i).indicator 1 | M] a‖ₑ ∂μ < ∞

Tactic:
[

Goal state after:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : CompleteSpace E
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ s ∈ C, MeasurableSet s
M : MeasurableSpace α := generateFrom C
Mle : M ≤ hα
μ' : Measure α := μ.trim Mle
hμ' : μ' = μ.trim Mle
m' : VectorMeasure α E
m'C : ∀ s ∈ C, m' s = m s
hm' : ∀ (s : Set α), ‖m' s‖ₑ ≤ μ' s
m'_le : m'.variation ≤ μ'
this✝ : MeasurableSpace α := hα
f : ℕ → Set α
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf : Pairwise (Function.onFun Disjoint f)
this :
  ∫ᵛ (x : α), μ[fun x ↦ ∑' (d : ℕ), (f d).indicator 1 x | M] x ∂•m' =
    ∫ᵛ (x : α), ∑' (d : ℕ), μ[(f d).indicator 1 | M] x ∂•m'
A : ∀ (i : ℕ), ∫⁻ (a : α), ‖μ[(f i).indicator 1 | M] a‖ₑ ∂μ = μ (f i)
⊢ ∑' (i : ℕ), ∫⁻ (a : α), ‖μ[(f i).indicator 1 | M] a‖ₑ ∂μ < ∞
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case pos.h_closed.hf
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
I : ∀ (f : ↥(Lp H 1 μ.variation)), ∫ᵛ (x : X), C (↑↑f x) ∂[B; μ] = ∫ᵛ (x : X), ↑↑(C.compLp f) x ∂[B; μ]
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun y ↦ ∫ᵛ (y_1 : X), C (↑↑y y_1) ∂[B; μ]

Tactic:
[

Goal state after:
case pos.h_closed.hf
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
I : ∀ (f : ↥(Lp H 1 μ.variation)), ∫ᵛ (x : X), C (↑↑f x) ∂[B; μ] = ∫ᵛ (x : X), ↑↑(C.compLp f) x ∂[B; μ]
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun y ↦ ∫ᵛ (y_1 : X), C (↑↑y y_1) ∂[B; μ]
Module: Mathlib.MeasureTheory.VectorMeasure.AddContent

Goal state before:
pp: α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : CompleteSpace E
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ s ∈ C, MeasurableSet s
M : MeasurableSpace α := generateFrom C
Mle : M ≤ hα
μ' : Measure α := μ.trim Mle
hμ' : μ' = μ.trim Mle
m' : VectorMeasure α E
m'C : ∀ s ∈ C, m' s = m s
hm' : ∀ (s : Set α), ‖m' s‖ₑ ≤ μ' s
m'_le : m'.variation ≤ μ'
this✝ : MeasurableSpace α := hα
f : ℕ → Set α
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf : Pairwise (Function.onFun Disjoint f)
this :
  ∫ᵛ (x : α), μ[fun x ↦ ∑' (d : ℕ), (f d).indicator 1 x | M] x ∂•m' =
    ∫ᵛ (x : α), ∑' (d : ℕ), μ[(f d).indicator 1 | M] x ∂•m'
I : ∑' (i : ℕ), ∫⁻ (a : α), ‖μ[(f i).indicator 1 | M] a‖ₑ ∂μ < ∞
A : ∀ (i : ℕ), Measurable fun x ↦ ‖μ[(f i).indicator 1 | M] x‖ₑ
⊢ ∑' (i : ℕ), ∫⁻ (a : α), ‖μ[(f i).indicator 1 | M] a‖ₑ ∂μ.trim Mle < ∞

Tactic:
[

Goal state after:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : CompleteSpace E
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ s ∈ C, MeasurableSet s
M : MeasurableSpace α := generateFrom C
Mle : M ≤ hα
μ' : Measure α := μ.trim Mle
hμ' : μ' = μ.trim Mle
m' : VectorMeasure α E
m'C : ∀ s ∈ C, m' s = m s
hm' : ∀ (s : Set α), ‖m' s‖ₑ ≤ μ' s
m'_le : m'.variation ≤ μ'
this✝ : MeasurableSpace α := hα
f : ℕ → Set α
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf : Pairwise (Function.onFun Disjoint f)
this :
  ∫ᵛ (x : α), μ[fun x ↦ ∑' (d : ℕ), (f d).indicator 1 x | M] x ∂•m' =
    ∫ᵛ (x : α), ∑' (d : ℕ), μ[(f d).indicator 1 | M] x ∂•m'
I : ∑' (i : ℕ), ∫⁻ (a : α), ‖μ[(f i).indicator 1 | M] a‖ₑ ∂μ < ∞
A : ∀ (i : ℕ), Measurable fun x ↦ ‖μ[(f i).indicator 1 | M] x‖ₑ
⊢ ∑' (i : ℕ), ∫⁻ (a : α), ‖μ[(f i).indicator 1 | M] a‖ₑ ∂μ.trim Mle < ∞
Module: Mathlib.MeasureTheory.VectorMeasure.Prod

Goal state before:
pp: X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : CompleteSpace G
inst✝ : IsFiniteMeasure μ.variation
s : Set (X × Y)
hs : MeasurableSet s
this : μ.prod ν B = μ.prodOfIsFiniteMeasureLeft ν B
⊢ (μ.prod ν B) s = ∫ᵛ (x : X), ν (Prod.mk x ⁻¹' s) ∂[B.flip; μ]

Tactic:
this

Goal state after:
X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : CompleteSpace G
inst✝ : IsFiniteMeasure μ.variation
s : Set (X × Y)
hs : MeasurableSet s
this : μ.prod ν B = μ.prodOfIsFiniteMeasureLeft ν B
⊢ (μ.prodOfIsFiniteMeasureLeft ν B) s = ∫ᵛ (x : X), ν (Prod.mk x ⁻¹' s) ∂[B.flip; μ]
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✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
s : ℕ → Set X
hm : ∀ (i : ℕ), MeasurableSet (s i)
hd : Pairwise (Disjoint on s)
hfi : μ.IntegrableOn f (⋃ i, s i)
hG : CompleteSpace G
I : ∑' (i : ℕ), ‖B‖ₑ * ∫⁻ (x : X) in s i, ‖f x‖ₑ ∂μ.variation < ∞
this : Summable fun n ↦ ∫ᵛ (x : X) in s n, f x ∂[B; μ]
⊢ Tendsto (fun n ↦ ∑ i ∈ Finset.range n, ∫ᵛ (x : X) in s i, f x ∂[B; μ]) atTop (𝓝 ∫ᵛ (x : X) in ⋃ n, s n, f x ∂[B; μ])

Tactic:
[

Goal state after:
case pos
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
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
s : ℕ → Set X
hm : ∀ (i : ℕ), MeasurableSet (s i)
hd : Pairwise (Disjoint on s)
hfi : μ.IntegrableOn f (⋃ i, s i)
hG : CompleteSpace G
I : ∑' (i : ℕ), ‖B‖ₑ * ∫⁻ (x : X) in s i, ‖f x‖ₑ ∂μ.variation < ∞
this : Summable fun n ↦ ∫ᵛ (x : X) in s n, f x ∂[B; μ]
⊢ Tendsto (fun n ↦ ∑ i ∈ Finset.range n, ∫ᵛ (x : X) in s i, f x ∂[B; μ]) atTop (𝓝 ∫ᵛ (x : X) in ⋃ n, s n, f x ∂[B; μ])
Module: Mathlib.MeasureTheory.VectorMeasure.Integral

Goal state before:
pp: case a.inr
X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : Nontrivial E
C : ℝ≥0
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = C * ‖x‖₊ * ‖y‖₊
hC : C ≠ 0
⊢ C • μ.variation ≤ (μ.transpose B).variation

Tactic:
suffices μ.variation ≤ C⁻¹ • (μ.transpose B).variation by
      grw [this, smul_smul, mul_inv_cancel₀ hC, one_smul]

Goal state after:
case a.inr
X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : Nontrivial E
C : ℝ≥0
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = C * ‖x‖₊ * ‖y‖₊
hC : C ≠ 0
⊢ μ.variation ≤ C⁻¹ • (μ.transpose B).variation