-- Mathlib/Analysis/SpecialFunctions/Pow/Integral.lean
theorem lintegral_rpow_eq_lintegral_meas_lt_mul :
    ∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
      ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (t ^ (p - 1))

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
⊢ ENNReal.ofReal p * ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) =
    ENNReal.ofReal p * ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))

Tactic:
apply congr_arg fun z => ENNReal.ofReal p * z

Result:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))

Proof state:
case h
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
⊢ (fun a => μ {a_1 | a ≤ f a_1} * ENNReal.ofReal (a ^ (p - 1))) =ᶠ[ae (volume.restrict (Ioi 0))] fun a =>
    μ {a_1 | a < f a_1} * ENNReal.ofReal (a ^ (p - 1))

Tactic:
filter_upwards [meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f]
  with t ht

Result:
case h
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
t : ℝ
ht : μ {a | t ≤ f a} = μ {a | t < f a}
⊢ μ {a | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) = μ {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω ^ p) ∂μ =
    ENNReal.ofReal p * ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))

Tactic:
rw [lintegral_rpow_eq_lintegral_meas_le_mul μ f_nn f_mble p_pos]

Result:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
⊢ ENNReal.ofReal p * ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) =
    ENNReal.ofReal p * ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))

Proof state:
case h
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
t : ℝ
ht : μ {a | t ≤ f a} = μ {a | t < f a}
⊢ μ {a | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) = μ {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))

Tactic:
rw [ht]

Result:
no goals

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))

Tactic:
apply lintegral_congr_ae

Result:
case h
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
⊢ (fun a => μ {a_1 | a ≤ f a_1} * ENNReal.ofReal (a ^ (p - 1))) =ᶠ[ae (volume.restrict (Ioi 0))] fun a =>
    μ {a_1 | a < f a_1} * ENNReal.ofReal (a ^ (p - 1))