Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: case a.inr
X : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
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
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝ : CompleteSpace G
hf : μ.Integrable f
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = ‖B.flip y‖₊ * ‖x‖₊
s : Set X
hs : MeasurableSet s
ε : NNReal
εpos : 0 < ε
δ : NNReal := ε / 3
δpos : 0 < δ
g : X →ₛ E
hg : eLpNorm (f - ⇑g) 1 (μ.transpose B).variation ≤ ↑δ
gmem : MemLp (⇑g) 1 μ.variation
I1 : ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation + ↑δ
I2 :
  ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation =
    ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i})
ρ : ℝ≥0∞
ρpos : ρ > 0
hρ : ∑ i ∈ g.range, ‖i‖ₑ * ρ ≤ ↑δ
P : E → Finset (Set X)
Pg : ∀ (i : E), ∀ t ∈ P i, t ⊆ ⇑g ⁻¹' {i}
Pdisj : ∀ (i : E), (↑(P i)).PairwiseDisjoint id
Pmeas : ∀ (i : E), ∀ t ∈ P i, MeasurableSet t
hP :
  ∀ (i : E),
    ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤
      ‖i‖ₑ * (∑ p ∈ P i, ‖((μ.transpose B).restrict s) p‖ₑ + ρ)
I3 :
  ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤
    ∑ i ∈ g.range.sigma P, ‖i.fst‖ₑ * ‖((μ.transpose B).restrict s) i.snd‖ₑ + ↑δ
i : E
p : Set X
h'i : i ≠ 0
hi : i ∈ g.range ∧ p ∈ P i
pmeas : MeasurableSet p
⊢ ‖⟨i, p⟩.fst‖ₑ * ‖((μ.transpose B).restrict s) ⟨i, p⟩.snd‖ₑ =
    ‖∫ᵛ (x : X) in ⟨i, p⟩.snd, ⟨i, p⟩.fst ∂[B; μ.restrict s]‖ₑ

Tactic:
have : IsFiniteMeasure ((μ.restrict s).variation.restrict p) := by
        constructor
        rw [variation_restrict hs, Measure.restrict_restrict pmeas,
          MeasureTheory.Measure.restrict_apply_univ]
        apply lt_of_le_of_lt ?_ (g.integrable_iff.1 (memLp_one_iff_integrable.1 gmem) i h'i)
        exact measure_mono (inter_subset_left.trans (Pg i _ hi.2))

Goal state after:
case a.inr
X : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
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
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝ : CompleteSpace G
hf : μ.Integrable f
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = ‖B.flip y‖₊ * ‖x‖₊
s : Set X
hs : MeasurableSet s
ε : NNReal
εpos : 0 < ε
δ : NNReal := ε / 3
δpos : 0 < δ
g : X →ₛ E
hg : eLpNorm (f - ⇑g) 1 (μ.transpose B).variation ≤ ↑δ
gmem : MemLp (⇑g) 1 μ.variation
I1 : ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation + ↑δ
I2 :
  ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation =
    ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i})
ρ : ℝ≥0∞
ρpos : ρ > 0
hρ : ∑ i ∈ g.range, ‖i‖ₑ * ρ ≤ ↑δ
P : E → Finset (Set X)
Pg : ∀ (i : E), ∀ t ∈ P i, t ⊆ ⇑g ⁻¹' {i}
Pdisj : ∀ (i : E), (↑(P i)).PairwiseDisjoint id
Pmeas : ∀ (i : E), ∀ t ∈ P i, MeasurableSet t
hP :
  ∀ (i : E),
    ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤
      ‖i‖ₑ * (∑ p ∈ P i, ‖((μ.transpose B).restrict s) p‖ₑ + ρ)
I3 :
  ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤
    ∑ i ∈ g.range.sigma P, ‖i.fst‖ₑ * ‖((μ.transpose B).restrict s) i.snd‖ₑ + ↑δ
i : E
p : Set X
h'i : i ≠ 0
hi : i ∈ g.range ∧ p ∈ P i
pmeas : MeasurableSet p
this : IsFiniteMeasure ((μ.restrict s).variation.restrict p)
⊢ ‖⟨i, p⟩.fst‖ₑ * ‖((μ.transpose B).restrict s) ⟨i, p⟩.snd‖ₑ =
    ‖∫ᵛ (x : X) in ⟨i, p⟩.snd, ⟨i, p⟩.fst ∂[B; μ.restrict s]‖ₑ