/-- Convolution is associative. This requires that
* all maps are a.e. strongly measurable w.r.t one of the measures
* `f ⋆[L, ν] g` exists almost everywhere
* `‖g‖ ⋆[μ] ‖k‖` exists almost everywhere
* `‖f‖ ⋆[ν] (‖g‖ ⋆[μ] ‖k‖)` exists at `x₀` -/
theorem convolution_assoc (hL : ∀ (x : E) (y : E') (z : E''), L₂ (L x y) z = L₃ x (L₄ y z)) {x₀ : G}
    (hf : AEStronglyMeasurable f ν) (hg : AEStronglyMeasurable g μ) (hk : AEStronglyMeasurable k μ)
    (hfg : ∀ᵐ y ∂μ, ConvolutionExistsAt f g y L ν)
    (hgk : ∀ᵐ x ∂ν, ConvolutionExistsAt (fun x => ‖g x‖) (fun x => ‖k x‖) x (mul ℝ ℝ) μ)
    (hfgk :
      ConvolutionExistsAt (fun x => ‖f x‖) ((fun x => ‖g x‖) ⋆[mul ℝ ℝ, μ] fun x => ‖k x‖) x₀
        (mul ℝ ℝ) ν) :
    ((f ⋆[L, ν] g) ⋆[L₂, μ] k) x₀ = (f ⋆[L₃, ν] g ⋆[L₄, μ] k) x₀ := by
  refine' convolution_assoc' L L₂ L₃ L₄ hL hfg (hgk.mono fun x hx => hx.ofNorm L₄ hg hk) _
  -- the following is similar to `Integrable.convolution_integrand`
  have h_meas :
    AEStronglyMeasurable (uncurry fun x y => L₃ (f y) (L₄ (g x) (k (x₀ - y - x))))
      (μ.prod ν) := by
    refine' L₃.aestronglyMeasurable_comp₂ hf.snd _
    refine' L₄.aestronglyMeasurable_comp₂ hg.fst _
    refine' (hk.mono' _).comp_measurable ((measurable_const.sub measurable_snd).sub measurable_fst)
    refine' QuasiMeasurePreserving.absolutelyContinuous _
    refine' QuasiMeasurePreserving.prod_of_left
      ((measurable_const.sub measurable_snd).sub measurable_fst) (eventually_of_forall fun y => _)
    dsimp only
    exact quasiMeasurePreserving_sub_left_of_right_invariant μ _
  have h2_meas :
    AEStronglyMeasurable (fun y => ∫ x, ‖L₃ (f y) (L₄ (g x) (k (x₀ - y - x)))‖ ∂μ) ν :=
    h_meas.prod_swap.norm.integral_prod_right'
  have h3 : map (fun z : G × G => (z.1 - z.2, z.2)) (μ.prod ν) = μ.prod ν :=
    (measurePreserving_sub_prod μ ν).map_eq
  suffices Integrable (uncurry fun x y => L₃ (f y) (L₄ (g x) (k (x₀ - y - x)))) (μ.prod ν) by
    rw [← h3] at this
    convert this.comp_measurable (measurable_sub.prod_mk measurable_snd)
    ext ⟨x, y⟩
    simp (config := { unfoldPartialApp := true }) only [uncurry, Function.comp_apply,
      sub_sub_sub_cancel_right]
  simp_rw [integrable_prod_iff' h_meas]
  refine' ⟨((quasiMeasurePreserving_sub_left_of_right_invariant ν x₀).ae hgk).mono fun t ht =>
    (L₃ (f t)).integrable_comp <| ht.ofNorm L₄ hg hk, _⟩
  refine' (hfgk.const_mul (‖L₃‖ * ‖L₄‖)).mono' h2_meas
    (((quasiMeasurePreserving_sub_left_of_right_invariant ν x₀).ae hgk).mono fun t ht => _)
  · simp_rw [convolution_def, mul_apply', mul_mul_mul_comm ‖L₃‖ ‖L₄‖, ← integral_mul_left]
    rw [Real.norm_of_nonneg]
    · refine' integral_mono_of_nonneg (eventually_of_forall fun t => norm_nonneg _)
        ((ht.const_mul _).const_mul _) (eventually_of_forall fun s => _)
      simp only [← mul_assoc ‖L₄‖]
      apply_rules [ContinuousLinearMap.le_of_op_norm₂_le_of_le, le_rfl]
    

Proof state:
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
P : Type uP
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f f' : G → E
g g' : G → E'
x x' : G
y y' : E
inst✝²² : IsROrC 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : NormedSpace 𝕜 E'
inst✝¹⁹ : NormedSpace 𝕜 E''
inst✝¹⁸ : NormedSpace ℝ F
inst✝¹⁷ : NormedSpace 𝕜 F
n : ℕ∞
inst✝¹⁶ : CompleteSpace F
inst✝¹⁵ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : CompleteSpace F'
inst✝¹⁰ : NormedAddCommGroup F''
inst✝⁹ : NormedSpace ℝ F''
inst✝⁸ : NormedSpace 𝕜 F''
inst✝⁷ : CompleteSpace F''
k : G → E''
L₂ : F →L[𝕜] E'' →L[𝕜] F'
L₃ : E →L[𝕜] F'' →L[𝕜] F'
L₄ : E' →L[𝕜] E'' →L[𝕜] F''
inst✝⁶ : AddGroup G
inst✝⁵ : SigmaFinite μ
inst✝⁴ : SigmaFinite ν
inst✝³ : IsAddRightInvariant μ
inst✝² : MeasurableAdd₂ G
inst✝¹ : IsAddRightInvariant ν
inst✝ : MeasurableNeg G
hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)
x₀ : G
hf : AEStronglyMeasurable f ν
hg : AEStronglyMeasurable g μ
hk : AEStronglyMeasurable k μ
hfg : ∀ᵐ (y : G) ∂μ, ConvolutionExistsAt f g y L
hgk : ∀ᵐ (x : G) ∂ν, ConvolutionExistsAt (fun x => ‖g x‖) (fun x => ‖k x‖) x (mul ℝ ℝ)
hfgk : ConvolutionExistsAt (fun x => ‖f x‖) (convolution (fun x => ‖g x‖) (fun x => ‖k x‖) (mul ℝ ℝ)) x₀ (mul ℝ ℝ)
h_meas : AEStronglyMeasurable (uncurry fun x y => (L₃ (f y)) ((L₄ (g x)) (k (x₀ - y - x)))) (Measure.prod μ ν)
h2_meas : AEStronglyMeasurable (fun y => ∫ (x : G), ‖(L₃ (f y)) ((L₄ (g x)) (k (x₀ - y - x)))‖ ∂μ) ν
h3 : Measure.map (fun z => (z.1 - z.2, z.2)) (Measure.prod μ ν) = Measure.prod μ ν
t : G
ht : ConvolutionExistsAt (fun x => ‖g x‖) (fun x => ‖k x‖) (x₀ - t) (mul ℝ ℝ)
⊢ 0 ≤ ∫ (x : G), ‖uncurry (fun x y => (L₃ (f y)) ((L₄ (g x)) (k (x₀ - y - x)))) (x, t)‖ ∂μ

Next tactic:
exact integral_nonneg fun x => norm_nonneg _