Theorem: MeasureTheory.lintegral_eq_lintegral_meas_le
File: Mathlib/MeasureTheory/Integral/Layercake.lean

Proof steps:
state_before: 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 μ
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
state_after: 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 μ
cst : ℝ → ℝ := fun x => 1
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
tactic: set cst := fun _ : ℝ => (1 : ℝ)
state_before: 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 μ
cst : ℝ → ℝ := fun x => 1
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
state_after: 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 μ
cst : ℝ → ℝ := fun x => 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
tactic: have cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t := fun _ _ =>
    intervalIntegrable_const
state_before: 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 μ
cst : ℝ → ℝ := fun x => 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
state_after: 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 μ
cst : ℝ → ℝ := fun x => 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
key :
  ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, cst t) ∂μ =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (cst t)
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
tactic: have key :=
    lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble cst_intble
      (eventually_of_forall fun _ => zero_le_one)
state_before: 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 μ
cst : ℝ → ℝ := fun x => 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
key :
  ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, cst t) ∂μ =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (cst t)
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
state_after: 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 μ
cst : ℝ → ℝ := fun x => 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
key : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, 1) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
tactic: simp_rw [cst, ENNReal.ofReal_one, mul_one] at key
state_before: 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 μ
cst : ℝ → ℝ := fun x => 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
key : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, 1) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
state_after: 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 μ
cst : ℝ → ℝ := fun x => 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
key : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, 1) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, 1) ∂μ
tactic: rw [← key]
state_before: 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 μ
cst : ℝ → ℝ := fun x => 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
key : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, 1) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, 1) ∂μ
state_after: case e_f.h
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 μ
cst : ℝ → ℝ := fun x => 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
key : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, 1) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
ω : α
⊢ ENNReal.ofReal (f ω) = ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, 1)
tactic: congr with ω
state_before: case e_f.h
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 μ
cst : ℝ → ℝ := fun x => 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
key : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, 1) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
ω : α
⊢ ENNReal.ofReal (f ω) = ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, 1)
state_after: no goals
tactic: simp only [intervalIntegral.integral_const, sub_zero, Algebra.id.smul_eq_mul, mul_one]