mathlib-initiative/mathlib-tactics huggingface dataset:
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)
⊢ HasSum (fun i ↦ if MeasurableSet (f i) then ∫ᵛ (x : X), ν (Prod.mk x ⁻¹' f i) ∂[B.flip; μ] else 0)
    (if MeasurableSet (⋃ i, f i) then ∫ᵛ (x : X), ν (Prod.mk x ⁻¹' ⋃ i, f i) ∂[B.flip; μ] else 0)

Tactic:
by

Goal state after:
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✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
inst✝⁹ : NormedAddCommGroup G
inst✝⁸ : NormedSpace ℝ G
inst✝⁷ : NormedAddCommGroup H
inst✝⁶ : NormedSpace ℝ H
inst✝⁵ : NormedAddCommGroup I
inst✝⁴ : NormedSpace ℝ I
inst✝³ : NormedAddCommGroup J
inst✝² : NormedSpace ℝ J
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : CompleteSpace G
inst✝ : IsFiniteMeasure μ.variation
⊢ ∃ ρ, ∀ (s : Set X) (t : Set Y), MeasurableSet s → MeasurableSet t → ρ (s ×ˢ t) = (B (μ s)) (ν t)

Tactic:
by

Goal state after:
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 < ∞
i : ℕ
⊢ Measurable fun a ↦ ‖μ[(f i).indicator 1 | M] a‖ₑ

Tactic:
by

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
C : ℝ≥0∞
hC : ∀ x ∈ s, ‖f x‖ₑ ≤ C
⊢ ‖∫ᵛ (x : X) in s, f x ∂[B; μ]‖ₑ ≤ C * ‖B‖ₑ * μ.variation s

Tactic:
by

Goal state after:
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✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
ρ : VectorMeasure (X × Y) G
hρ : ∀ (s : Set X) (t : Set Y), MeasurableSet s → MeasurableSet t → ρ (s ×ˢ t) = (B (μ s)) (ν t)
⊢ μ.prod ν B = ρ

Tactic:
by

Goal state after:
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)
⊢ HasSum (fun i ↦ if MeasurableSet (f i) then ∫ᵛ (x : α), μ[(f i).indicator 1 | M] x ∂•m' else 0)
    (if MeasurableSet (⋃ i, f i) then ∫ᵛ (x : α), μ[(⋃ i, f i).indicator 1 | M] x ∂•m' else 0)

Tactic:
by

Goal state after:
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
⊢ μ.prod ν B = μ.prodOfIsFiniteMeasureLeft ν B

Tactic:
by

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
C : ℝ
h : IsFiniteMeasure (μ.variation.restrict s)
hC : ∀ᵐ (x : X) ∂μ.variation.restrict s, ‖f x‖ ≤ C
hs : ¬MeasurableSet s
h's : ¬μ.variation s = 0
⊢ (ae (μ.variation.restrict s)).NeBot

Tactic:
by

Goal state after:
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
⊢ (μ.prod ν B) s = ∫ᵛ (x : X), ν (Prod.mk x ⁻¹' s) ∂[B.flip; μ]

Tactic:
by

Goal state after:
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
inst✝¹ : CompleteSpace G
inst✝ : IsFiniteMeasure μ.variation
B : F →L[ℝ] E →L[ℝ] G
s : Set (X × Y)
hs : MeasurableSet s
⊢ (μ.prod ν B.flip) s = ∫ᵛ (x : X), ν (Prod.mk x ⁻¹' s) ∂[B; μ]

Tactic:
by

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
C : ℝ
h : IsFiniteMeasure (μ.variation.restrict s)
hC : ∀ᵐ (x : X) ∂μ.variation.restrict s, ‖f x‖ ≤ C
⊢ ‖∫ᵛ (x : X) in s, f x ∂[B; μ]‖ ≤ C * ‖B‖ * μ.variation.real s

Tactic:
by

Goal state after:
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✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
inst✝² : CompleteSpace G
inst✝¹ : IsFiniteMeasure μ.variation
inst✝ : SFinite ν.variation
⊢ (μ.prod ν B).variation ≤ ‖B‖ₑ • μ.variation.prod ν.variation

Tactic:
by

Goal state after:
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✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace ℝ E
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace ℝ F
inst✝¹⁰ : NormedAddCommGroup G
inst✝⁹ : NormedSpace ℝ G
inst✝⁸ : NormedAddCommGroup H
inst✝⁷ : NormedSpace ℝ H
inst✝⁶ : NormedAddCommGroup I
inst✝⁵ : NormedSpace ℝ I
inst✝⁴ : NormedAddCommGroup J
inst✝³ : NormedSpace ℝ J
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
inst✝² : CompleteSpace G
inst✝¹ : IsFiniteMeasure μ.variation
inst✝ : IsFiniteMeasure ν.variation
⊢ IsFiniteMeasure (‖B‖ₑ • μ.variation.prod ν.variation)

Tactic:
by

Goal state after:
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✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace ℝ E
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace ℝ F
inst✝¹⁰ : NormedAddCommGroup G
inst✝⁹ : NormedSpace ℝ G
inst✝⁸ : NormedAddCommGroup H
inst✝⁷ : NormedSpace ℝ H
inst✝⁶ : NormedAddCommGroup I
inst✝⁵ : NormedSpace ℝ I
inst✝⁴ : NormedAddCommGroup J
inst✝³ : NormedSpace ℝ J
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
inst✝² : CompleteSpace G
inst✝¹ : IsFiniteMeasure μ.variation
inst✝ : IsFiniteMeasure ν.variation
⊢ IsFiniteMeasure (μ.prod ν B).variation

Tactic:
by

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
s : SignedMeasure X
i : Set X
⊢ ‖s i‖ ≤ s.totalVariation.real i

Tactic:
by

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
s : SignedMeasure X
i : Set X
⊢ ENNReal.ofReal (s.totalVariation.real i) = s.totalVariation i

Tactic:
by

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : SignedMeasure X
i j : Set X
him : MeasurableSet i
hi : 0 ≤[i] μ
hjm : MeasurableSet j
⊢ (μ.toMeasureOfZeroLE i him hi) j = ‖μ (i ∩ j)‖ₑ

Tactic:
by

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : SignedMeasure X
i j : Set X
him : MeasurableSet i
hi : μ ≤[i] 0
hjm : MeasurableSet j
⊢ (μ.toMeasureOfLEZero i him hi) j = ‖μ (i ∩ j)‖ₑ

Tactic:
by

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
C : ℝ
h : IsFiniteMeasure (μ.variation.restrict s)
hC : ∀ x ∈ s, ‖f x‖ ≤ C
⊢ ‖∫ᵛ (x : X) in s, f x ∂[B; μ]‖ ≤ C * ‖B‖ * μ.variation.real s

Tactic:
by

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : SignedMeasure X
r : Set X
hr : MeasurableSet r
s : Set X
hs : MeasurableSet s
hpos : 0 ≤[s] μ
hneg : μ ≤[sᶜ] 0
hposPart : μ.toJordanDecomposition.posPart = μ.toMeasureOfZeroLE s hs hpos
hnegPart : μ.toJordanDecomposition.negPart = μ.toMeasureOfLEZero sᶜ ⋯ hneg
⊢ μ.totalVariation r = ‖μ (s ∩ r)‖ₑ + ‖μ (sᶜ ∩ r)‖ₑ

Tactic:
by

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
⊢ ‖∫ᵛ (x : X) in s, f x ∂[B; μ]‖ₑ ≤ ‖B‖ₑ * ∫⁻ (x : X) in s, ‖f x‖ₑ ∂μ.variation

Tactic:
by

Goal state after:
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
m'' : VectorMeasure α E :=
  { measureOf' := fun s ↦ if MeasurableSet s then ∫ᵛ (x : α), μ[s.indicator 1 | M] x ∂•m' else 0, empty' := ⋯,
    not_measurable' := ⋯, m_iUnion' := ⋯ }
s : Set α
hs : s ∈ C
⊢ ∫ᵛ (x : α), μ[s.indicator 1 | M] x ∂•m' = ∫ᵛ (x : α), s.indicator 1 x ∂•m'

Tactic:
by

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : SignedMeasure X
r : Set X
hr : MeasurableSet r
s : Set X
hs : MeasurableSet s
hpos : 0 ≤[s] μ
hneg : μ ≤[sᶜ] 0
hposPart : μ.toJordanDecomposition.posPart = μ.toMeasureOfZeroLE s hs hpos
hnegPart : μ.toJordanDecomposition.negPart = μ.toMeasureOfLEZero sᶜ ⋯ hneg
⊢ Disjoint (s ∩ r) (sᶜ ∩ r)

Tactic:
by

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : SignedMeasure X
r : Set X
hr : MeasurableSet r
s : Set X
hs : MeasurableSet s
hpos : 0 ≤[s] μ
hneg : μ ≤[sᶜ] 0
hposPart : μ.toJordanDecomposition.posPart = μ.toMeasureOfZeroLE s hs hpos
hnegPart : μ.toJordanDecomposition.negPart = μ.toMeasureOfLEZero sᶜ ⋯ hneg
⊢ (VectorMeasure.variation μ) (s ∩ r ∪ sᶜ ∩ r) = (VectorMeasure.variation μ) r

Tactic:
by

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
⊢ ‖∫ᵛ (x : X) in s, f x ∂[B; μ]‖ₑ ≤ ∫⁻ (x : X) in s, ‖f x‖ₑ ∂(μ.transpose B).variation

Tactic:
by

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : SignedMeasure X
⊢ μ.totalVariation = VectorMeasure.variation μ

Tactic:
by

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

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
⊢ ⇑(μ.transpose B) = cbmApplyMeasure μ B

Tactic:
by

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

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
s : Set X
x : E
⊢ (cbmApplyMeasure μ B s) x = (B x) (μ s)

Tactic:
by

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
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 : ℕ), ‖B‖ₑ * ∫⁻ (x : X) in s i, ‖f x‖ₑ ∂μ.variation = ‖B‖ₑ * ∫⁻ (x : X) in ⋃ i, s i, ‖f x‖ₑ ∂μ.variation

Tactic:
by

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

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
s t : Set X
hs : MeasurableSet s
ht : MeasurableSet t
hdisj : Disjoint s t
⊢ cbmApplyMeasure μ B (s ∪ t) = cbmApplyMeasure μ B s + cbmApplyMeasure μ B t

Tactic:
by

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

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
s : Set X
⊢ 0 ≤ ‖B‖ * ‖μ s‖

Tactic:
by

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

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
s : Set X
⊢ ‖cbmApplyMeasure μ B s‖ ≤ ‖B‖ * ‖μ s‖

Tactic:
rw [opNorm_le_iff (by positivity)]

Goal state after:
X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
s : Set X
⊢ ∀ (x : E), ‖(cbmApplyMeasure μ B s) x‖ ≤ ‖B‖ * ‖μ s‖ * ‖x‖
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
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)
hG : CompleteSpace G
hfi : Integrable f (μ.variation.restrict (⋃ i, s i))
⊢ ‖B‖ₑ < ∞

Tactic:
by

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
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
⊢ ‖B‖ₑ * ∫⁻ (x : X) in ⋃ i, s i, ‖f x‖ₑ ∂μ.variation < ∞

Tactic:
by

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

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
s : Set X
⊢ ‖cbmApplyMeasure μ B s‖ ≤ ‖B‖ * ‖μ s‖

Tactic:
by

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

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
⊢ DominatedFinMeasAdditive μ.variation ⇑(μ.transpose B) ‖B‖

Tactic:
by

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
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 < ∞
⊢ Summable fun n ↦ ∫ᵛ (x : X) in s n, f x ∂[B; μ]

Tactic:
by

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

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
⊢ DominatedFinMeasAdditive (μ.transpose B).variation (⇑(μ.transpose B)) 1

Tactic:
by

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

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
⊢ transpose 0 B = 0

Tactic:
by

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

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
s : Set X
⊢ (μ.restrict s).transpose B = (μ.transpose B).restrict s

Tactic:
by

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

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
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
φ : X → Y
⊢ (μ.map φ).transpose B = (μ.transpose B).map φ

Tactic:
by

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

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ ν : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
⊢ (μ + ν).transpose B = μ.transpose B + ν.transpose B

Tactic:
by

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

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
c : ℝ
⊢ (c • μ).transpose B = c • μ.transpose B

Tactic:
by

Goal state after: