α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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 ≤ᵐ[μ] g
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
h : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ
⊢ f ≤ᵐ[μ] g
have A :
  ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ε ≤ f x ∧ g x ≤ N} ∩ spanningSets μ p) = 0 := by
  intro ε N p εpos
  let s := {x | g x + ε ≤ f x ∧ g x ≤ N} ∩ spanningSets μ p
  have s_meas : MeasurableSet s := by
    have A : MeasurableSet {x | g x + ε ≤ f x} := measurableSet_le (hg.add measurable_const) hf
    have B : MeasurableSet {x | g x ≤ N} := measurableSet_le hg measurable_const
    exact (A.inter B).inter (measurable_spanningSets μ p)
  have s_lt_top : μ s < ∞ :=
    (measure_mono (Set.inter_subset_right _ _)).trans_lt (measure_spanningSets_lt_top μ p)
  have A : (∫⁻ x in s, g x ∂μ) + ε * μ s ≤ (∫⁻ x in s, g x ∂μ) + 0 :=
    calc
      (∫⁻ x in s, g x ∂μ) + ε * μ s = (∫⁻ x in s, g x ∂μ) + ∫⁻ _ in s, ε ∂μ := by
        simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]
      _ = ∫⁻ x in s, g x + ε ∂μ := (lintegral_add_right _ measurable_const).symm
      _ ≤ ∫⁻ x in s, f x ∂μ :=
        (set_lintegral_mono (hg.add measurable_const) hf fun x hx => hx.1.1)
      _ ≤ (∫⁻ x in s, g x ∂μ) + 0 := by rw [add_zero]; exact h s s_meas s_lt_top
  have B : (∫⁻ x in s, g x ∂μ) ≠ ∞ := by
    apply ne_of_lt
    calc
      (∫⁻ x in s, g x ∂μ) ≤ ∫⁻ _ in s, N ∂μ :=
        set_lintegral_mono hg measurable_const fun x hx => hx.1.2
      _ = N * μ s := by
        simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]
      _ < ∞ := by
        simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,
          ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff]
  have : (ε : ℝ≥0∞) * μ s ≤ 0 := ENNReal.le_of_add_le_add_left B A
  simpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, εpos.ne', false_or_iff]
case intro.intro.intro
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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 ≤ᵐ[μ] g
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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 ≤ᵐ[μ] g
obtain ⟨u, _, u_pos, u_lim⟩ :
  ∃ u : ℕ → ℝ≥0, StrictAnti u ∧ (∀ n, 0 < u n) ∧ Tendsto u atTop (nhds 0) :=
  exists_seq_strictAnti_tendsto (0 : ℝ≥0)
case intro.intro.intro
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
⊢ f ≤ᵐ[μ] g
case intro.intro.intro
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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 ≤ᵐ[μ] g
let s := fun n : ℕ => {x | g x + u n ≤ f x ∧ g x ≤ (n : ℝ≥0)} ∩ spanningSets μ n
case intro.intro.intro
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
⊢ f ≤ᵐ[μ] g
case intro.intro.intro
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
⊢ f ≤ᵐ[μ] g
have μs : ∀ n, μ (s n) = 0 := fun n => A _ _ _ (u_pos n)
case intro.intro.intro
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
B : {x | f x ≤ g x}ᶜ ⊆ ⋃ (n : ℕ), s n
⊢ f ≤ᵐ[μ] g
case intro.intro.intro
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
⊢ f ≤ᵐ[μ] g
have B : {x | f x ≤ g x}ᶜ ⊆ ⋃ n, s n := by
  intro x hx
  simp at hx
  have L1 : ∀ᶠ n in atTop, g x + u n ≤ f x := by
    have : Tendsto (fun n => g x + u n) atTop (𝓝 (g x + (0 : ℝ≥0))) :=
      tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)
    simp at this
    exact eventually_le_of_tendsto_lt hx this
  have L2 : ∀ᶠ n : ℕ in (atTop : Filter ℕ), g x ≤ (n : ℝ≥0) :=
    haveI : Tendsto (fun n : ℕ => ((n : ℝ≥0) : ℝ≥0∞)) atTop (𝓝 ∞) := by
      simp only [ENNReal.coe_nat]
      exact ENNReal.tendsto_nat_nhds_top
    eventually_ge_of_tendsto_gt (hx.trans_le le_top) this
  apply Set.mem_iUnion.2
  exact ((L1.and L2).and (eventually_mem_spanningSets μ x)).exists
case intro.intro.intro
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
B : {x | f x ≤ g x}ᶜ ⊆ ⋃ (n : ℕ), s n
⊢ ↑↑μ ({x | (fun x => f x ≤ g x) x}ᶜ) ≤ 0
case intro.intro.intro
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
B : {x | f x ≤ g x}ᶜ ⊆ ⋃ (n : ℕ), s n
⊢ f ≤ᵐ[μ] g
refine' le_antisymm _ bot_le
no goals
case intro.intro.intro
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
B : {x | f x ≤ g x}ᶜ ⊆ ⋃ (n : ℕ), s n
⊢ ↑↑μ ({x | (fun x => f x ≤ g x) x}ᶜ) ≤ 0
calc
  μ ({x : α | (fun x : α => f x ≤ g x) x}ᶜ) ≤ μ (⋃ n, s n) := measure_mono B
  _ ≤ ∑' n, μ (s n) := (measure_iUnion_le _)
  _ = 0 := by simp only [μs, tsum_zero]
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
intro ε N p εpos
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
let s := {x | g x + ε ≤ f x ∧ g x ≤ N} ∩ spanningSets μ p
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
have s_meas : MeasurableSet s := by
  have A : MeasurableSet {x | g x + ε ≤ f x} := measurableSet_le (hg.add measurable_const) hf
  have B : MeasurableSet {x | g x ≤ N} := measurableSet_le hg measurable_const
  exact (A.inter B).inter (measurable_spanningSets μ p)
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
have s_lt_top : μ s < ∞ :=
  (measure_mono (Set.inter_subset_right _ _)).trans_lt (measure_spanningSets_lt_top μ p)
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
A : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0
⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
have A : (∫⁻ x in s, g x ∂μ) + ε * μ s ≤ (∫⁻ x in s, g x ∂μ) + 0 :=
  calc
    (∫⁻ x in s, g x ∂μ) + ε * μ s = (∫⁻ x in s, g x ∂μ) + ∫⁻ _ in s, ε ∂μ := by
      simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]
    _ = ∫⁻ x in s, g x + ε ∂μ := (lintegral_add_right _ measurable_const).symm
    _ ≤ ∫⁻ x in s, f x ∂μ :=
      (set_lintegral_mono (hg.add measurable_const) hf fun x hx => hx.1.1)
    _ ≤ (∫⁻ x in s, g x ∂μ) + 0 := by rw [add_zero]; exact h s s_meas s_lt_top
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
A : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0
B : (∫⁻ (x : α) in s, g x ∂μ) ≠ ⊤
⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
A : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0
⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
have B : (∫⁻ x in s, g x ∂μ) ≠ ∞ := by
  apply ne_of_lt
  calc
    (∫⁻ x in s, g x ∂μ) ≤ ∫⁻ _ in s, N ∂μ :=
      set_lintegral_mono hg measurable_const fun x hx => hx.1.2
    _ = N * μ s := by
      simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]
    _ < ∞ := by
      simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,
        ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff]
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
A : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0
B : (∫⁻ (x : α) in s, g x ∂μ) ≠ ⊤
this : ↑ε * ↑↑μ s ≤ 0
⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
A : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0
B : (∫⁻ (x : α) in s, g x ∂μ) ≠ ⊤
⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
have : (ε : ℝ≥0∞) * μ s ≤ 0 := ENNReal.le_of_add_le_add_left B A
no goals
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
A : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0
B : (∫⁻ (x : α) in s, g x ∂μ) ≠ ⊤
this : ↑ε * ↑↑μ s ≤ 0
⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
simpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, εpos.ne', false_or_iff]
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
A : MeasurableSet {x | g x + ↑ε ≤ f x}
⊢ MeasurableSet s
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
⊢ MeasurableSet s
have A : MeasurableSet {x | g x + ε ≤ f x} := measurableSet_le (hg.add measurable_const) hf
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
A : MeasurableSet {x | g x + ↑ε ≤ f x}
B : MeasurableSet {x | g x ≤ ↑N}
⊢ MeasurableSet s
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
A : MeasurableSet {x | g x + ↑ε ≤ f x}
⊢ MeasurableSet s
have B : MeasurableSet {x | g x ≤ N} := measurableSet_le hg measurable_const
no goals
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
A : MeasurableSet {x | g x + ↑ε ≤ f x}
B : MeasurableSet {x | g x ≤ ↑N}
⊢ MeasurableSet s
exact (A.inter B).inter (measurable_spanningSets μ p)
no goals
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
⊢ (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s = (∫⁻ (x : α) in s, g x ∂μ) + ∫⁻ (x : α) in s, ↑ε ∂μ
simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
⊢ (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
⊢ (∫⁻ (x : α) in s, f x ∂μ) ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0
rw [add_zero]
no goals
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
⊢ (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ
exact h s s_meas s_lt_top
case h
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
A : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0
⊢ (∫⁻ (x : α) in s, g x ∂μ) < ⊤
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
A : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0
⊢ (∫⁻ (x : α) in s, g x ∂μ) ≠ ⊤
apply ne_of_lt
no goals
case h
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
A : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0
⊢ (∫⁻ (x : α) in s, g x ∂μ) < ⊤
calc
  (∫⁻ x in s, g x ∂μ) ≤ ∫⁻ _ in s, N ∂μ :=
    set_lintegral_mono hg measurable_const fun x hx => hx.1.2
  _ = N * μ s := by
    simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]
  _ < ∞ := by
    simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,
      ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff]
no goals
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
A : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0
⊢ (∫⁻ (x : α) in s, ↑N ∂μ) = ↑N * ↑↑μ s
simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]
no goals
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p✝ : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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_meas : MeasurableSet s
s_lt_top : ↑↑μ s < ⊤
A : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0
⊢ ↑N * ↑↑μ s < ⊤
simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,
  ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff]
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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 : x ∈ {x | f x ≤ g x}ᶜ
⊢ x ∈ ⋃ (n : ℕ), s n
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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 | f x ≤ g x}ᶜ ⊆ ⋃ (n : ℕ), s n
intro x hx
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
⊢ x ∈ ⋃ (n : ℕ), s n
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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 : x ∈ {x | f x ≤ g x}ᶜ
⊢ x ∈ ⋃ (n : ℕ), s n
simp at hx
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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 : ℕ), s n
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
⊢ x ∈ ⋃ (n : ℕ), s n
have L1 : ∀ᶠ n in atTop, g x + u n ≤ f x := by
  have : Tendsto (fun n => g x + u n) atTop (𝓝 (g x + (0 : ℝ≥0))) :=
    tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)
  simp at this
  exact eventually_le_of_tendsto_lt hx this
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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 : ℕ), s n
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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 : ℕ), s n
have L2 : ∀ᶠ n : ℕ in (atTop : Filter ℕ), g x ≤ (n : ℝ≥0) :=
  haveI : Tendsto (fun n : ℕ => ((n : ℝ≥0) : ℝ≥0∞)) atTop (𝓝 ∞) := by
    simp only [ENNReal.coe_nat]
    exact ENNReal.tendsto_nat_nhds_top
  eventually_ge_of_tendsto_gt (hx.trans_le le_top) this
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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 : ℕ), s n
apply Set.mem_iUnion.2
no goals
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
exact ((L1.and L2).and (eventually_mem_spanningSets μ x)).exists
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
⊢ ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x
have : Tendsto (fun n => g x + u n) atTop (𝓝 (g x + (0 : ℝ≥0))) :=
  tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
simp at this
no goals
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
exact eventually_le_of_tendsto_lt hx this
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
⊢ Tendsto (fun n => ↑n) atTop (𝓝 ⊤)
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
⊢ Tendsto (fun n => ↑↑n) atTop (𝓝 ⊤)
simp only [ENNReal.coe_nat]
no goals
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
⊢ Tendsto (fun n => ↑n) atTop (𝓝 ⊤)
exact ENNReal.tendsto_nat_nhds_top
no goals
α : Type u_1
E : Type ?u.48454
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
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
B : {x | f x ≤ g x}ᶜ ⊆ ⋃ (n : ℕ), s n
⊢ (∑' (n : ℕ), ↑↑μ (s n)) = 0
simp only [μs, tsum_zero]