-- Mathlib/MeasureTheory/Function/AEEqOfLIntegral.lean
theorem ae_le_of_forall_setLIntegral_le_of_sigmaFinite₀ [SigmaFinite μ]
    {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ) :
    f ≤ᵐ[μ] g

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
A : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
s : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n
μs : ∀ (n : ℕ), μ (s n) = 0
x : α
hx : g x < f x
L1 : ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x
⊢ x ∈ ⋃ n, s n

Tactic:
have L2 : ∀ᶠ n : ℕ in (atTop : Filter ℕ), g x ≤ (n : ℝ≥0) :=
  have : Tendsto (fun n : ℕ => ((n : ℝ≥0) : ℝ≥0∞)) atTop (𝓝 ∞) := by
    simp only [ENNReal.coe_natCast]
    exact ENNReal.tendsto_nat_nhds_top
  this.eventually_const_le (hx.trans_le le_top)

Result:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
A : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
s : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n
μs : ∀ (n : ℕ), μ (s n) = 0
x : α
hx : g x < f x
L1 : ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x
L2 : ∀ᶠ (n : ℕ) in atTop, g x ≤ ↑↑n
⊢ x ∈ ⋃ n, s n

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
ε N : ℝ≥0
p : ℕ
εpos : 0 < ε
s : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p
s_lt_top : μ s < ⊤
⊢ μ (toMeasurable μ s) < ⊤

Tactic:
rwa [measure_toMeasurable]

Result:
no goals

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
A : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
s : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n
μs : ∀ (n : ℕ), μ (s n) = 0
x : α
hx : g x < f x
this : Tendsto (fun n => g x + ↑(u n)) atTop (𝓝 (g x + ↑0))
⊢ ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x

Tactic:
simp only [ENNReal.coe_zero, add_zero] at this

Result:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
A : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
s : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n
μs : ∀ (n : ℕ), μ (s n) = 0
x : α
hx : g x < f x
this : Tendsto (fun n => g x + ↑(u n)) atTop (𝓝 (g x))
⊢ ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
A : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
s : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n
μs : ∀ (n : ℕ), μ (s n) = 0
x : α
hx : g x < f x
L1 : ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x
L2 : ∀ᶠ (n : ℕ) in atTop, g x ≤ ↑↑n
⊢ ∃ i, x ∈ s i

Tactic:
exact ((L1.and L2).and (eventually_mem_spanningSets μ x)).exists

Result:
no goals

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
A : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
s : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n
μs : ∀ (n : ℕ), μ (s n) = 0
x : α
hx : g x < f x
L1 : ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x
L2 : ∀ᶠ (n : ℕ) in atTop, g x ≤ ↑↑n
⊢ x ∈ ⋃ n, s n

Tactic:
apply Set.mem_iUnion.2

Result:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
A : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
s : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n
μs : ∀ (n : ℕ), μ (s n) = 0
x : α
hx : g x < f x
L1 : ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x
L2 : ∀ᶠ (n : ℕ) in atTop, g x ≤ ↑↑n
⊢ ∃ i, x ∈ s i

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
A : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
⊢ f ≤ᶠ[ae μ] g

Tactic:
obtain ⟨u, _, u_pos, u_lim⟩ :
  ∃ u : ℕ → ℝ≥0, StrictAnti u ∧ (∀ n, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
  exists_seq_strictAnti_tendsto (0 : ℝ≥0)

Result:
case intro.intro.intro
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
A : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
⊢ f ≤ᶠ[ae μ] g

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
⊢ ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0

Tactic:
intro ε N p εpos

Result:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
ε N : ℝ≥0
p : ℕ
εpos : 0 < ε
⊢ μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
A : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
s : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n
μs : ∀ (n : ℕ), μ (s n) = 0
x : α
hx : g x < f x
this : Tendsto (fun n => g x + ↑(u n)) atTop (𝓝 (g x))
⊢ ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x

Tactic:
exact this.eventually_le_const hx

Result:
no goals