/-- The layer cake formula / **Cavalieri's principle** / tail probability formula:

Let `f` be a non-negative measurable function on a measure space. Let `G` be an
increasing absolutely continuous function on the positive real line, vanishing at the origin,
with derivative `G' = g`. Then the integral of the composition `G ∘ f` can be written as
the integral over the positive real line of the "tail measures" `μ {ω | f(ω) ≥ t}` of `f`
weighted by `g`.

Roughly speaking, the statement is: `∫⁻ (G ∘ f) ∂μ = ∫⁻ t in 0..∞, g(t) * μ {ω | f(ω) ≥ t}`.

See `lintegral_comp_eq_lintegral_meas_lt_mul` for a version with sets of the form `{ω | f(ω) > t}`
instead. -/
theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
    (f_mble : AEMeasurable f μ) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
    (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 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
  obtain ⟨G, G_mble, G_nn, g_eq_G⟩ : ∃ G : ℝ → ℝ, Measurable G ∧ 0 ≤ G
      ∧ g =ᵐ[volume.restrict (Ioi 0)] G := by
    refine' AEMeasurable.exists_measurable_nonneg _ g_nn
    exact aemeasurable_Ioi_of_forall_Ioc fun t ht => (g_intble t ht).1.1.aemeasurable
  have g_eq_G_on : ∀ t, g =ᵐ[volume.restrict (Ioc 0 t)] G := fun t =>
    ae_mono (Measure.restrict_mono Ioc_subset_Ioi_self le_rfl) g_eq_G
  have G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t := by
    refine' fun t t_pos => ⟨(g_intble t t_pos).1.congr_fun_ae (g_eq_G_on t), _⟩
    rw [Ioc_eq_empty_of_le t_pos.lt.le]
    exact integrableOn_empty
  obtain ⟨F, F_mble, F_nn, f_eq_F⟩ : ∃ F : α → ℝ, Measurable F ∧ 0 ≤ F ∧ f =ᵐ[μ] F := by
    refine ⟨fun ω ↦ max (f_mble.mk f ω) 0, f_mble.measurable_mk.max measurable_const,
        fun ω ↦ le_max_right _ _, ?_⟩
    filter_upwards [f_mble.ae_eq_mk, f_nn] with ω hω h'ω
    rw [← hω]
    exact (max_eq_left h'ω).symm
  have eq₁ :
    (∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t)) =
      ∫⁻ t in Ioi 0, μ {a : α | t ≤ F a} * ENNReal.ofReal (G t) := by
    apply lintegral_congr_ae
    filter_upwards [g_eq_G] with t ht
    rw [ht]
    congr 1
    apply measure_congr
    filter_upwards [f_eq_F] with a ha using by simp [setOf, ha]
  have eq₂ : ∀ᵐ ω ∂μ,
      ENNReal.ofReal (∫ t in (0)..f ω, g t) = ENNReal.ofReal (∫ t in (0)..F ω, G t) := by
    filter_upwards [f_eq_F] with ω fω_nn
    rw [fω_nn]
    congr 1
    refine' intervalIntegral.integral_congr_ae _
    have fω_nn : 0 ≤ F ω := F_nn ω
    rw [uIoc_of_le fω_nn, ←
      ae_restrict_iff' (measurableSet_Ioc : MeasurableSet (Ioc (0 : ℝ) (F ω)))]
    exact g_eq_G_on (F ω)
  

Proof state:
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
s : Set α
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂Measure.restrict volume (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᶠ[ae (Measure.restrict volume (Ioi 0))] G
g_eq_G_on : ∀ (t : ℝ), g =ᶠ[ae (Measure.restrict volume (Ioc 0 t))] G
G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t
F : α → ℝ
F_mble : Measurable F
F_nn : 0 ≤ F
f_eq_F : f =ᶠ[ae μ] F
eq₁ :
  ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ F a} * ENNReal.ofReal (G t)
eq₂ : ∀ᵐ (ω : α) ∂μ, ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ENNReal.ofReal (∫ (t : ℝ) in 0 ..F ω, G t)
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) ∂μ =
    ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)

Next tactic:
simp_rw [lintegral_congr_ae eq₂, eq₁]