-- Mathlib/MeasureTheory/Integral/Layercake.lean
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)

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
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 (volume.restrict (Ioi 0))] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0 ..s, g t → μ {a | s < f a} ≠ ⊤
M_bdd : BddAbove {s | g =ᶠ[ae (volume.restrict (Ioc 0 s))] 0}
M : ℝ := sSup {s | g =ᶠ[ae (volume.restrict (Ioc 0 s))] 0}
zero_mem : 0 ∈ {s | g =ᶠ[ae (volume.restrict (Ioc 0 s))] 0}
M_nonneg : 0 ≤ M
hgM : g =ᶠ[ae (volume.restrict (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : ν {a | f a ≤ M} = 0
J : μ {a | u n < f a} < ⊤
⊢ ν {a | u n < f a} ≤ μ {a | u n < f a}

Tactic:
exact restrict_le_self _

Result:
no goals

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
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 (volume.restrict (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}
⊢ ENNReal.ofReal (∫ (t : ℝ) in 0 ..s, g t) ≤ ENNReal.ofReal (∫ (t : ℝ) in 0 ..f a, g t)

Tactic:
apply ENNReal.ofReal_le_ofReal

Result:
case h
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
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 (volume.restrict (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

Proof state:
case pos.intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
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 (volume.restrict (Ioi 0))] 0
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (t : ℝ) in 0 ..s, g t
h's : μ {a | s < f a} = ⊤
⊢ ∫⁻ (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)

Tactic:
rw [intervalIntegral.integral_of_le s_pos.le] at hs

Result:
case pos.intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
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 (volume.restrict (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} = ⊤
⊢ ∫⁻ (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)

Proof state:
case hf
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
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 (volume.restrict (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} = ⊤
I_pos : ∫⁻ (a : ℝ) in Ioc 0 s, ENNReal.ofReal (g a) ≠ 0
⊢ Measurable fun t => ENNReal.ofReal (g t)

Tactic:
exact ENNReal.measurable_ofReal.comp g_mble

Result:
no goals

Proof state:
case h
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
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 (volume.restrict (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

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

Result:
case h.hf
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
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 (volume.restrict (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}
⊢ 0 ≤ᶠ[ae (volume.restrict (Ioc 0 (f a)))] g

case h.hfi
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
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 (volume.restrict (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}
⊢ IntervalIntegrable g volume 0 (f a)

Proof state:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
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 (volume.restrict (Ioi 0))] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0 ..s, g t → μ {a | s < f a} ≠ ⊤
M_bdd : BddAbove {s | g =ᶠ[ae (volume.restrict (Ioc 0 s))] 0}
M : ℝ := sSup {s | g =ᶠ[ae (volume.restrict (Ioc 0 s))] 0}
zero_mem : 0 ∈ {s | g =ᶠ[ae (volume.restrict (Ioc 0 s))] 0}
M_nonneg : 0 ≤ M
hgM : g =ᶠ[ae (volume.restrict (Ioc 0 M))] 0
⊢ ∫⁻ (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)

Tactic:
let ν := μ.restrict {a : α | M < f a}

Result:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
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 (volume.restrict (Ioi 0))] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0 ..s, g t → μ {a | s < f a} ≠ ⊤
M_bdd : BddAbove {s | g =ᶠ[ae (volume.restrict (Ioc 0 s))] 0}
M : ℝ := sSup {s | g =ᶠ[ae (volume.restrict (Ioc 0 s))] 0}
zero_mem : 0 ∈ {s | g =ᶠ[ae (volume.restrict (Ioc 0 s))] 0}
M_nonneg : 0 ≤ M
hgM : g =ᶠ[ae (volume.restrict (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a | M < f a}
⊢ ∫⁻ (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)