/-- An auxiliary version of the layer cake formula (Cavalieri's principle, tail probability
formula), with a measurability assumption that would also essentially follow from the
integrability assumptions, and a sigma-finiteness assumption.

See `MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul` and
`lintegral_comp_eq_lintegral_meas_lt_mul` for the main formulations of the layer
cake formula. -/
theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
    (μ : Measure α) [SigmaFinite μ]
    (f_nn : 0 ≤ f) (f_mble : Measurable f)
    (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
    (g_nn : ∀ t > 0, 0 ≤ g t) :
    (∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ) =
      ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) := by
  have g_intble' : ∀ t : ℝ, 0 ≤ t → IntervalIntegrable g volume 0 t := by
    intro t ht
    cases' eq_or_lt_of_le ht with h h
    · simp [← h]
    · exact g_intble t h
  have integrand_eq : ∀ ω,
      ENNReal.ofReal (∫ t in (0)..f ω, g t) = (∫⁻ t in Ioc 0 (f ω), ENNReal.ofReal (g t)) := by
    intro ω
    have g_ae_nn : 0 ≤ᵐ[volume.restrict (Ioc 0 (f ω))] g := by
      filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f ω)))]
        with x hx using g_nn x hx.1
    rw [← ofReal_integral_eq_lintegral_ofReal (g_intble' (f ω) (f_nn ω)).1 g_ae_nn]
    congr
    exact intervalIntegral.integral_of_le (f_nn ω)
  rw [lintegral_congr integrand_eq]
  simp_rw [← lintegral_indicator (fun t => ENNReal.ofReal (g t)) measurableSet_Ioc]
  -- Porting note: was part of `simp_rw` on the previous line, but didn't trigger.
  rw [← lintegral_indicator _ measurableSet_Ioi, lintegral_lintegral_swap]
  · apply congr_arg
    funext s
    have aux₁ :
      (fun x => (Ioc 0 (f x)).indicator (fun t : ℝ => ENNReal.ofReal (g t)) s) = fun x =>
        ENNReal.ofReal (g s) * (Ioi (0 : ℝ)).indicator (fun _ => 1) s *
          (Ici s).indicator (fun _ : ℝ => (1 : ℝ≥0∞)) (f x) := by
      funext a
      by_cases h : s ∈ Ioc (0 : ℝ) (f a)
      · simp only [h, show s ∈ Ioi (0 : ℝ) from h.1, show f a ∈ Ici s from h.2, indicator_of_mem,
          mul_one]
      · have h_copy := h
        simp only [mem_Ioc, not_and, not_le] at h
        by_cases h' : 0 < s
        · simp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le, mul_zero]
        · have : s ∉ Ioi (0 : ℝ) := h'
          simp only [this, h', indicator_of_not_mem, not_false_iff, mul_zero,
            zero_mul, mem_Ioc, false_and_iff]
    

Proof state:
case h.h
α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
s✝ : Set α
μ : Measure α
inst✝ : SigmaFinite μ
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t
integrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
s : ℝ
aux₁ :
  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>
    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)
⊢ ∫⁻ (x : α), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s ∂μ =
    indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s

Next tactic:
simp_rw [aux₁]