case intro
α : Type u_2
β : Type ?u.48
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_1
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (i : ι), Measurable (f i)
lim : Tendsto f u (𝓝 g)
x : ℕ → ι
hx : Tendsto x atTop u
⊢ Measurable g
α : Type u_2
β : Type ?u.48
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_1
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (i : ι), Measurable (f i)
lim : Tendsto f u (𝓝 g)
⊢ Measurable g
rcases u.exists_seq_tendsto with ⟨x, hx⟩
case intro
α : Type u_2
β : Type ?u.48
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_1
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (i : ι), Measurable (f i)
lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x))
x : ℕ → ι
hx : Tendsto x atTop u
⊢ Measurable g
case intro
α : Type u_2
β : Type ?u.48
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_1
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (i : ι), Measurable (f i)
lim : Tendsto f u (𝓝 g)
x : ℕ → ι
hx : Tendsto x atTop u
⊢ Measurable g
rw [tendsto_pi_nhds] at lim
case intro
α : Type u_2
β : Type ?u.48
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_1
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (i : ι), Measurable (f i)
lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x))
x : ℕ → ι
hx : Tendsto x atTop u
this : (fun y => liminf (fun n => f (x n) y) atTop) = g
⊢ Measurable g
case intro
α : Type u_2
β : Type ?u.48
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_1
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (i : ι), Measurable (f i)
lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x))
x : ℕ → ι
hx : Tendsto x atTop u
⊢ Measurable g
have : (fun y => liminf (fun n => (f (x n) y : ℝ≥0∞)) atTop) = g := by
  ext1 y
  exact ((lim y).comp hx).liminf_eq
case intro
α : Type u_2
β : Type ?u.48
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_1
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (i : ι), Measurable (f i)
lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x))
x : ℕ → ι
hx : Tendsto x atTop u
this : (fun y => liminf (fun n => f (x n) y) atTop) = g
⊢ Measurable fun y => liminf (fun n => f (x n) y) atTop
case intro
α : Type u_2
β : Type ?u.48
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_1
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (i : ι), Measurable (f i)
lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x))
x : ℕ → ι
hx : Tendsto x atTop u
this : (fun y => liminf (fun n => f (x n) y) atTop) = g
⊢ Measurable g
rw [← this]
case intro
α : Type u_2
β : Type ?u.48
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_1
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (i : ι), Measurable (f i)
lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x))
x : ℕ → ι
hx : Tendsto x atTop u
this : (fun y => liminf (fun n => f (x n) y) atTop) = g
⊢ Measurable fun y => liminf (fun n => f (x n) y) atTop
case intro
α : Type u_2
β : Type ?u.48
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_1
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (i : ι), Measurable (f i)
lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x))
x : ℕ → ι
hx : Tendsto x atTop u
this : (fun y => liminf (fun n => f (x n) y) atTop) = g
⊢ Measurable fun y => liminf (fun n => f (x n) y) atTop
show Measurable fun y => liminf (fun n => (f (x n) y : ℝ≥0∞)) atTop
no goals
case intro
α : Type u_2
β : Type ?u.48
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_1
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (i : ι), Measurable (f i)
lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x))
x : ℕ → ι
hx : Tendsto x atTop u
this : (fun y => liminf (fun n => f (x n) y) atTop) = g
⊢ Measurable fun y => liminf (fun n => f (x n) y) atTop
exact measurable_liminf fun n => hf (x n)
case h
α : Type u_2
β : Type ?u.48
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_1
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (i : ι), Measurable (f i)
lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x))
x : ℕ → ι
hx : Tendsto x atTop u
y : α
⊢ liminf (fun n => f (x n) y) atTop = g y
α : Type u_2
β : Type ?u.48
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_1
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (i : ι), Measurable (f i)
lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x))
x : ℕ → ι
hx : Tendsto x atTop u
⊢ (fun y => liminf (fun n => f (x n) y) atTop) = g
ext1 y
no goals
case h
α : Type u_2
β : Type ?u.48
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_1
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (i : ι), Measurable (f i)
lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x))
x : ℕ → ι
hx : Tendsto x atTop u
y : α
⊢ liminf (fun n => f (x n) y) atTop = g y
exact ((lim y).comp hx).liminf_eq