Theorem: ENNReal.measurable_of_tendsto'
File: Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean

tactic: rcases u.exists_seq_tendsto with ⟨x, hx⟩
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u✝ : Set α
inst✝² : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), Measurable (f i)
lim : Tendsto f u (𝓝 g)
⊢ Measurable g
state_after: case intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u✝ : Set α
inst✝² : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), Measurable (f i)
lim : Tendsto f u (𝓝 g)
x : ℕ → ι
hx : Tendsto x atTop u
⊢ Measurable g

tactic: rw [tendsto_pi_nhds] at lim
state_before: case intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u✝ : Set α
inst✝² : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), Measurable (f i)
lim : Tendsto f u (𝓝 g)
x : ℕ → ι
hx : Tendsto x atTop u
⊢ Measurable g
state_after: case intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u✝ : Set α
inst✝² : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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

tactic: have : (fun y => liminf (fun n => (f (x n) y : ℝ≥0∞)) atTop) = g := by
  ext1 y
  exact ((lim y).comp hx).liminf_eq
state_before: case intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u✝ : Set α
inst✝² : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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
state_after: case intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u✝ : Set α
inst✝² : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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

tactic: rw [← this]
state_before: case intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u✝ : Set α
inst✝² : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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
state_after: case intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u✝ : Set α
inst✝² : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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

tactic: exact measurable_liminf fun n => hf (x n)
state_before: case intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u✝ : Set α
inst✝² : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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
state_after: no goals

tactic: ext1 y
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u✝ : Set α
inst✝² : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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
state_after: case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u✝ : Set α
inst✝² : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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

tactic: exact ((lim y).comp hx).liminf_eq
state_before: case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u✝ : Set α
inst✝² : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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
state_after: no goals