Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: 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)
⊢ ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ (μ.withDensity f B).variation s + ↑ε

Tactic:
have I3 : ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (g ⁻¹' {i}) ≤
      ∑ i ∈ g.range.sigma P, ‖i.1‖ₑ * ‖(μ.transpose B).restrict s i.2‖ₑ + δ := calc
    ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (g ⁻¹' {i})
    _ ≤ ∑ i ∈ g.range, ‖i‖ₑ * ((∑ p ∈ P i, ‖(μ.transpose B).restrict s p‖ₑ) + ρ) := by
      gcongr 1 with i hi
      exact hP i
    _ ≤ ∑ i ∈ g.range, ∑ p ∈ P i, ‖i‖ₑ * ‖(μ.transpose B).restrict s p‖ₑ + δ := by
      simp_rw [mul_add, Finset.sum_add_distrib, Finset.mul_sum]
      gcongr
    _ = ∑ i ∈ g.range.sigma P, ‖i.1‖ₑ * ‖(μ.transpose B).restrict s i.2‖ₑ + δ := by
      rw [Finset.sum_sigma']

Goal state after:
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
⊢ ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ (μ.withDensity f B).variation s + ↑ε