/-- 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.
Compared to `lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite`, we remove
the sigma-finite 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 (μ : Measure α)
    (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
  /- We will reduce to the sigma-finite case, after excluding two easy cases where the result
  is more or less obvious. -/
  have f_nonneg : ∀ ω, 0 ≤ f ω := fun ω ↦ f_nn ω
  -- trivial case where `g` is ae zero. Then both integrals vanish.
  by_cases H1 : g =ᵐ[volume.restrict (Ioi (0 : ℝ))] 0
  · have A : ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ = 0 := by
      have : ∀ ω, ∫ t in (0)..f ω, g t = ∫ t in (0)..f ω, 0 := by
        intro ω
        simp_rw [intervalIntegral.integral_of_le (f_nonneg ω)]
        apply integral_congr_ae
        exact ae_restrict_of_ae_restrict_of_subset Ioc_subset_Ioi_self H1
      simp [this]
    have B : ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) = 0 := by
      have : (fun t ↦ μ {a : α | t ≤ f a} * ENNReal.ofReal (g t))
        =ᵐ[volume.restrict (Ioi (0:ℝ))] 0 := by
          filter_upwards [H1] with t ht using by simp [ht]
      simp [lintegral_congr_ae this]
    rw [A, B]
  -- easy case where both sides are obviously infinite: for some `s`, one has
  -- `μ {a : α | s < f a} = ∞` and moreover `g` is not ae zero on `[0, s]`.
  by_cases H2 : ∃ s > 0, 0 < ∫ t in (0)..s, g t ∧ μ {a : α | s < f a} = ∞
  · rcases H2 with ⟨s, s_pos, hs, h's⟩
    rw [intervalIntegral.integral_of_le s_pos.le] at hs
    /- The first integral is infinite, as for `t ∈ [0, s]` one has `μ {a : α | t ≤ f a} = ∞`,
    and moreover the additional integral `g` is not uniformly zero. -/
    have A : ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) = ∞ := by
      rw [eq_top_iff]
      calc
      ∞ = ∫⁻ t in Ioc 0 s, ∞ * ENNReal.ofReal (g t) := by
          have I_pos : ∫⁻ (a : ℝ) in Ioc 0 s, ENNReal.ofReal (g a) ≠ 0 := by
            rw [← ofReal_integral_eq_lintegral_ofReal (g_intble s s_pos).1]
            · simpa only [not_lt, ge_iff_le, ne_eq, ENNReal.ofReal_eq_zero, not_le] using hs
            · filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht using g_nn _ ht.1
          rw [lintegral_const_mul, ENNReal.top_mul I_pos]
          exact ENNReal.measurable_ofReal.comp g_mble
      _ ≤ ∫⁻ t in Ioc 0 s, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) := by
          apply set_lintegral_mono' measurableSet_Ioc (fun x hx ↦ ?_)
          rw [← h's]
          gcongr
          exact fun a ha ↦ hx.2.trans (le_of_lt ha)
      _ ≤ ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) :=
          lintegral_mono_set Ioc_subset_Ioi_self
    /- The second integral is infinite, as one integrates amont other things on those `ω` where
    `f ω > s`: this is an infinite measure set, and on it the integrand is bounded below
    by `∫ t in 0..s, g t` which is positive. -/
    have B : ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ = ∞ := by
      rw [eq_top_iff]
      calc
      ∞ = ∫⁻ _ in {a | s < f a}, ENNReal.ofReal (∫ t in (0)..s, g t) ∂μ := by
          simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter,
            h's, ne_eq, ENNReal.ofReal_eq_zero, not_le]
          rw [ENNReal.mul_top]
          simpa [intervalIntegral.integral_of_le s_pos.le] using hs
      _ ≤ ∫⁻ ω in {a | s < f a}, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ := by
          apply set_lintegral_mono' (measurableSet_lt measurable_const f_mble) (fun a ha ↦ ?_)
          apply ENNReal.ofReal_le_ofReal
          

Proof state:
case h
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
s✝ : Set α
μ : Measure α
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
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (Measure.restrict volume (Ioi 0))] 0
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (x : ℝ) in Ioc 0 s, g x ∂volume
h's : ↑↑μ {a | s < f a} = ⊤
A : ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t) = ⊤
a : α
ha : a ∈ {a | s < f a}
⊢ ∫ (t : ℝ) in 0 ..s, g t ≤ ∫ (t : ℝ) in 0 ..f a, g t

Next tactic:
apply intervalIntegral.integral_mono_interval le_rfl s_pos.le (le_of_lt ha)