case h
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
⊢ Tendsto (fun a => ⨍ (y : α) in a, f y ∂μ) (filterAt v x) (𝓝 (f x))
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ⨍ (y : α) in a, f y ∂μ) (filterAt v x) (𝓝 (f x))
filter_upwards [v.ae_tendsto_average_norm_sub hf, v.ae_eventually_measure_pos] with x hx h'x
case h
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
⊢ Tendsto (fun e => ‖(⨍ (y : α) in e, f y ∂μ) - f x‖) (filterAt v x) (𝓝 0)
case h
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
⊢ Tendsto (fun a => ⨍ (y : α) in a, f y ∂μ) (filterAt v x) (𝓝 (f x))
rw [tendsto_iff_norm_tendsto_zero]
case h
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
⊢ ∀ᶠ (t : Set α) in filterAt v x, ‖(⨍ (y : α) in t, f y ∂μ) - f x‖ ≤ ⨍ (y : α) in t, ‖f y - f x‖ ∂μ
case h
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
⊢ Tendsto (fun e => ‖(⨍ (y : α) in e, f y ∂μ) - f x‖) (filterAt v x) (𝓝 0)
refine' squeeze_zero' (eventually_of_forall fun a => norm_nonneg _) _ hx
case h
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
a : Set α
ha : 0 < ↑↑μ a
h'a : ↑↑μ a < ⊤
⊢ ‖(⨍ (y : α) in a, f y ∂μ) - f x‖ ≤ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ
case h
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
⊢ ∀ᶠ (t : Set α) in filterAt v x, ‖(⨍ (y : α) in t, f y ∂μ) - f x‖ ≤ ⨍ (y : α) in t, ‖f y - f x‖ ∂μ
filter_upwards [h'x, v.eventually_measure_lt_top x] with a ha h'a
case h
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
a : Set α
ha : 0 < ↑↑μ a
h'a : ↑↑μ a < ⊤
⊢ ‖(⨍ (y : α) in a, f y ∂μ) - ⨍ (x_1 : α) in a, f x ∂μ‖ ≤ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ
case h
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
a : Set α
ha : 0 < ↑↑μ a
h'a : ↑↑μ a < ⊤
⊢ ‖(⨍ (y : α) in a, f y ∂μ) - f x‖ ≤ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ
nth_rw 1 [← set_average_const ha.ne' h'a.ne (f x)]
case h
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
a : Set α
ha : 0 < ↑↑μ a
h'a : ↑↑μ a < ⊤
⊢ ‖(∫ (x : α), f x ∂(↑↑μ a)⁻¹ • Measure.restrict μ a) - ∫ (x_1 : α), f x ∂(↑↑μ a)⁻¹ • Measure.restrict μ a‖ ≤
    ∫ (x_1 : α), ‖f x_1 - f x‖ ∂(↑↑μ a)⁻¹ • Measure.restrict μ a
case h
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
a : Set α
ha : 0 < ↑↑μ a
h'a : ↑↑μ a < ⊤
⊢ ‖(⨍ (y : α) in a, f y ∂μ) - ⨍ (x_1 : α) in a, f x ∂μ‖ ≤ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ
simp_rw [set_average_eq']
case h
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
a : Set α
ha : 0 < ↑↑μ a
h'a : ↑↑μ a < ⊤
⊢ ‖∫ (a : α), f a - f x ∂(↑↑μ a)⁻¹ • Measure.restrict μ a‖ ≤
    ∫ (x_1 : α), ‖f x_1 - f x‖ ∂(↑↑μ a)⁻¹ • Measure.restrict μ a

case h.hf
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
a : Set α
ha : 0 < ↑↑μ a
h'a : ↑↑μ a < ⊤
⊢ Integrable fun x => f x

case h.hg
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
a : Set α
ha : 0 < ↑↑μ a
h'a : ↑↑μ a < ⊤
⊢ Integrable fun x_1 => f x
case h
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
a : Set α
ha : 0 < ↑↑μ a
h'a : ↑↑μ a < ⊤
⊢ ‖(∫ (x : α), f x ∂(↑↑μ a)⁻¹ • Measure.restrict μ a) - ∫ (x_1 : α), f x ∂(↑↑μ a)⁻¹ • Measure.restrict μ a‖ ≤
    ∫ (x_1 : α), ‖f x_1 - f x‖ ∂(↑↑μ a)⁻¹ • Measure.restrict μ a
rw [← integral_sub]
no goals
case h
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
a : Set α
ha : 0 < ↑↑μ a
h'a : ↑↑μ a < ⊤
⊢ ‖∫ (a : α), f a - f x ∂(↑↑μ a)⁻¹ • Measure.restrict μ a‖ ≤
    ∫ (x_1 : α), ‖f x_1 - f x‖ ∂(↑↑μ a)⁻¹ • Measure.restrict μ a
exact norm_integral_le_integral_norm _
no goals
case h.hf
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
a : Set α
ha : 0 < ↑↑μ a
h'a : ↑↑μ a < ⊤
⊢ Integrable fun x => f x
exact (integrable_inv_smul_measure ha.ne' h'a.ne).2 hf.integrableOn
no goals
case h.hg
α : Type u_2
inst✝⁷ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝² : IsLocallyFiniteMeasure ρ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f
x : α
hx : Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a
a : Set α
ha : 0 < ↑↑μ a
h'a : ↑↑μ a < ⊤
⊢ Integrable fun x_1 => f x
exact (integrable_inv_smul_measure ha.ne' h'a.ne).2 (integrableOn_const.2 (Or.inr h'a))