Theorem: MeasureTheory.Integrable.integral_eq_integral_meas_lt
File: Mathlib/MeasureTheory/Integral/Layercake.lean

tactic: have key := lintegral_eq_lintegral_meas_lt μ f_nn f_intble.aemeasurable
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
⊢ ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Ioi 0, (μ {a | t < f a}).toReal
state_after: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
⊢ ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Ioi 0, (μ {a | t < f a}).toReal

tactic: have lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ∞ := Integrable.lintegral_lt_top f_intble
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
⊢ ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Ioi 0, (μ {a | t < f a}).toReal
state_after: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
⊢ ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Ioi 0, (μ {a | t < f a}).toReal

tactic: have rhs_finite : ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t < f a} < ∞ := by simp only [← key, lhs_finite]
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
⊢ ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Ioi 0, (μ {a | t < f a}).toReal
state_after: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
⊢ ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Ioi 0, (μ {a | t < f a}).toReal

tactic: have rhs_integrand_finite : ∀ (t : ℝ), t > 0 → μ {a | t < f a} < ∞ :=
  fun t ht ↦ measure_gt_lt_top f_intble ht
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
⊢ ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Ioi 0, (μ {a | t < f a}).toReal
state_after: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
⊢ ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Ioi 0, (μ {a | t < f a}).toReal

tactic: convert (ENNReal.toReal_eq_toReal lhs_finite.ne rhs_finite.ne).mpr key
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
⊢ ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Ioi 0, (μ {a | t < f a}).toReal
state_after: case h.e'_2
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
⊢ ∫ (ω : α), f ω ∂μ = (∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ).toReal

case h.e'_3
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
⊢ ∫ (t : ℝ) in Ioi 0, (μ {a | t < f a}).toReal = (∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}).toReal

tactic: simp only [← key, lhs_finite]
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
state_after: no goals

tactic: exact integral_eq_lintegral_of_nonneg_ae f_nn f_intble.aestronglyMeasurable
state_before: case h.e'_2
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
⊢ ∫ (ω : α), f ω ∂μ = (∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ).toReal
state_after: no goals

tactic: have aux := @integral_eq_lintegral_of_nonneg_ae _ _ ((volume : Measure ℝ).restrict (Set.Ioi 0))
  (fun t ↦ ENNReal.toReal (μ {a : α | t < f a})) ?_ ?_
state_before: case h.e'_3
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
⊢ ∫ (t : ℝ) in Ioi 0, (μ {a | t < f a}).toReal = (∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}).toReal
state_after: case h.e'_3.refine_3
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
aux :
  ∫ (a : ℝ) in Ioi 0, (μ {a_1 | a < f a_1}).toReal =
    (∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ {a_1 | a < f a_1}).toReal).toReal
⊢ ∫ (t : ℝ) in Ioi 0, (μ {a | t < f a}).toReal = (∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}).toReal

case h.e'_3.refine_1
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
⊢ 0 ≤ᶠ[ae (volume.restrict (Ioi 0))] fun t => (μ {a | t < f a}).toReal

case h.e'_3.refine_2
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
⊢ AEStronglyMeasurable (fun t => (μ {a | t < f a}).toReal) (volume.restrict (Ioi 0))

tactic: rw [aux]
state_before: case h.e'_3.refine_3
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
aux :
  ∫ (a : ℝ) in Ioi 0, (μ {a_1 | a < f a_1}).toReal =
    (∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ {a_1 | a < f a_1}).toReal).toReal
⊢ ∫ (t : ℝ) in Ioi 0, (μ {a | t < f a}).toReal = (∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}).toReal
state_after: case h.e'_3.refine_3
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
aux :
  ∫ (a : ℝ) in Ioi 0, (μ {a_1 | a < f a_1}).toReal =
    (∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ {a_1 | a < f a_1}).toReal).toReal
⊢ (∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ {a_1 | a < f a_1}).toReal).toReal =
    (∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}).toReal

tactic: congr 1
state_before: case h.e'_3.refine_3
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
aux :
  ∫ (a : ℝ) in Ioi 0, (μ {a_1 | a < f a_1}).toReal =
    (∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ {a_1 | a < f a_1}).toReal).toReal
⊢ (∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ {a_1 | a < f a_1}).toReal).toReal =
    (∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}).toReal
state_after: case h.e'_3.refine_3.e_a
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
aux :
  ∫ (a : ℝ) in Ioi 0, (μ {a_1 | a < f a_1}).toReal =
    (∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ {a_1 | a < f a_1}).toReal).toReal
⊢ ∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ {a_1 | a < f a_1}).toReal = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}

tactic: apply setLIntegral_congr_fun measurableSet_Ioi (eventually_of_forall _)
state_before: case h.e'_3.refine_3.e_a
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
aux :
  ∫ (a : ℝ) in Ioi 0, (μ {a_1 | a < f a_1}).toReal =
    (∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ {a_1 | a < f a_1}).toReal).toReal
⊢ ∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ {a_1 | a < f a_1}).toReal = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
state_after: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
aux :
  ∫ (a : ℝ) in Ioi 0, (μ {a_1 | a < f a_1}).toReal =
    (∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ {a_1 | a < f a_1}).toReal).toReal
⊢ ∀ x ∈ Ioi 0, ENNReal.ofReal (μ {a | x < f a}).toReal = μ {a | x < f a}

tactic: exact fun t t_pos ↦ ENNReal.ofReal_toReal (rhs_integrand_finite t t_pos).ne
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
aux :
  ∫ (a : ℝ) in Ioi 0, (μ {a_1 | a < f a_1}).toReal =
    (∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ {a_1 | a < f a_1}).toReal).toReal
⊢ ∀ x ∈ Ioi 0, ENNReal.ofReal (μ {a | x < f a}).toReal = μ {a | x < f a}
state_after: no goals

tactic: exact eventually_of_forall (fun x ↦ by simp only [Pi.zero_apply, ENNReal.toReal_nonneg])
state_before: case h.e'_3.refine_1
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
⊢ 0 ≤ᶠ[ae (volume.restrict (Ioi 0))] fun t => (μ {a | t < f a}).toReal
state_after: no goals

tactic: simp only [Pi.zero_apply, ENNReal.toReal_nonneg]
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
x : ℝ
⊢ 0 x ≤ (fun t => (μ {a | t < f a}).toReal) x
state_after: no goals

tactic: apply Measurable.aestronglyMeasurable
state_before: case h.e'_3.refine_2
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
⊢ AEStronglyMeasurable (fun t => (μ {a | t < f a}).toReal) (volume.restrict (Ioi 0))
state_after: case h.e'_3.refine_2.hf
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
⊢ Measurable fun t => (μ {a | t < f a}).toReal

tactic: refine Measurable.ennreal_toReal ?_
state_before: case h.e'_3.refine_2.hf
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
⊢ Measurable fun t => (μ {a | t < f a}).toReal
state_after: case h.e'_3.refine_2.hf
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
⊢ Measurable fun t => μ {a | t < f a}

tactic: exact Antitone.measurable (fun _ _ hst ↦ measure_mono (fun _ h ↦ lt_of_le_of_lt hst h))
state_before: case h.e'_3.refine_2.hf
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤
⊢ Measurable fun t => μ {a | t < f a}
state_after: no goals