Theorem: MeasureTheory.measure_of_cont_bdd_of_tendsto_indicator
File: Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean

tactic: have fs_lim' :
  ∀ ω, Tendsto (fun n : ℕ ↦ (fs n ω : ℝ≥0)) atTop (𝓝 (indicator E (fun _ ↦ (1 : ℝ≥0)) ω)) := by
  rw [tendsto_pi_nhds] at fs_lim
  exact fun ω ↦ fs_lim ω
state_before: Ω : Type u_1
inst✝⁴ : TopologicalSpace Ω
inst✝³ : MeasurableSpace Ω
inst✝² inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ℕ → Ω →ᵇ ℝ≥0
fs_bdd : ∀ (n : ℕ) (ω : Ω), (fs n) ω ≤ c
fs_lim : Tendsto (fun n ω => (fs n) ω) atTop (𝓝 (E.indicator fun x => 1))
⊢ Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂μ) atTop (𝓝 (μ E))
state_after: Ω : Type u_1
inst✝⁴ : TopologicalSpace Ω
inst✝³ : MeasurableSpace Ω
inst✝² inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ℕ → Ω →ᵇ ℝ≥0
fs_bdd : ∀ (n : ℕ) (ω : Ω), (fs n) ω ≤ c
fs_lim : Tendsto (fun n ω => (fs n) ω) atTop (𝓝 (E.indicator fun x => 1))
fs_lim' : ∀ (ω : Ω), Tendsto (fun n => (fs n) ω) atTop (𝓝 (E.indicator (fun x => 1) ω))
⊢ Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂μ) atTop (𝓝 (μ E))

tactic: apply measure_of_cont_bdd_of_tendsto_filter_indicator μ E_mble fs
  (eventually_of_forall fun n ↦ eventually_of_forall (fs_bdd n)) (eventually_of_forall fs_lim')
state_before: Ω : Type u_1
inst✝⁴ : TopologicalSpace Ω
inst✝³ : MeasurableSpace Ω
inst✝² inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ℕ → Ω →ᵇ ℝ≥0
fs_bdd : ∀ (n : ℕ) (ω : Ω), (fs n) ω ≤ c
fs_lim : Tendsto (fun n ω => (fs n) ω) atTop (𝓝 (E.indicator fun x => 1))
fs_lim' : ∀ (ω : Ω), Tendsto (fun n => (fs n) ω) atTop (𝓝 (E.indicator (fun x => 1) ω))
⊢ Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂μ) atTop (𝓝 (μ E))
state_after: no goals

tactic: rw [tendsto_pi_nhds] at fs_lim
state_before: Ω : Type u_1
inst✝⁴ : TopologicalSpace Ω
inst✝³ : MeasurableSpace Ω
inst✝² inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ℕ → Ω →ᵇ ℝ≥0
fs_bdd : ∀ (n : ℕ) (ω : Ω), (fs n) ω ≤ c
fs_lim : Tendsto (fun n ω => (fs n) ω) atTop (𝓝 (E.indicator fun x => 1))
⊢ ∀ (ω : Ω), Tendsto (fun n => (fs n) ω) atTop (𝓝 (E.indicator (fun x => 1) ω))
state_after: Ω : Type u_1
inst✝⁴ : TopologicalSpace Ω
inst✝³ : MeasurableSpace Ω
inst✝² inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ℕ → Ω →ᵇ ℝ≥0
fs_bdd : ∀ (n : ℕ) (ω : Ω), (fs n) ω ≤ c
fs_lim : ∀ (x : Ω), Tendsto (fun i => (fs i) x) atTop (𝓝 (E.indicator (fun x => 1) x))
⊢ ∀ (ω : Ω), Tendsto (fun n => (fs n) ω) atTop (𝓝 (E.indicator (fun x => 1) ω))

tactic: exact fun ω ↦ fs_lim ω
state_before: Ω : Type u_1
inst✝⁴ : TopologicalSpace Ω
inst✝³ : MeasurableSpace Ω
inst✝² inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ℕ → Ω →ᵇ ℝ≥0
fs_bdd : ∀ (n : ℕ) (ω : Ω), (fs n) ω ≤ c
fs_lim : ∀ (x : Ω), Tendsto (fun i => (fs i) x) atTop (𝓝 (E.indicator (fun x => 1) x))
⊢ ∀ (ω : Ω), Tendsto (fun n => (fs n) ω) atTop (𝓝 (E.indicator (fun x => 1) ω))
state_after: no goals