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

tactic: convert tendsto_lintegral_nn_filter_of_le_const μ fs_bdd fs_lim
state_before: Ω : Type u_1
inst✝⁶ : TopologicalSpace Ω
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝³ : L.IsCountablyGenerated
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ι → Ω →ᵇ ℝ≥0
fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => (fs i) ω) L (𝓝 (E.indicator (fun x => 1) ω))
⊢ Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂μ) L (𝓝 (μ E))
state_after: case h.e'_5.h.e'_3
Ω : Type u_1
inst✝⁶ : TopologicalSpace Ω
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝³ : L.IsCountablyGenerated
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ι → Ω →ᵇ ℝ≥0
fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => (fs i) ω) L (𝓝 (E.indicator (fun x => 1) ω))
⊢ μ E = ∫⁻ (ω : Ω), ↑(E.indicator (fun x => 1) ω) ∂μ

tactic: have aux : ∀ ω, indicator E (fun _ ↦ (1 : ℝ≥0∞)) ω = ↑(indicator E (fun _ ↦ (1 : ℝ≥0)) ω) :=
  fun ω ↦ by simp only [ENNReal.coe_indicator, ENNReal.coe_one]
state_before: case h.e'_5.h.e'_3
Ω : Type u_1
inst✝⁶ : TopologicalSpace Ω
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝³ : L.IsCountablyGenerated
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ι → Ω →ᵇ ℝ≥0
fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => (fs i) ω) L (𝓝 (E.indicator (fun x => 1) ω))
⊢ μ E = ∫⁻ (ω : Ω), ↑(E.indicator (fun x => 1) ω) ∂μ
state_after: case h.e'_5.h.e'_3
Ω : Type u_1
inst✝⁶ : TopologicalSpace Ω
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝³ : L.IsCountablyGenerated
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ι → Ω →ᵇ ℝ≥0
fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => (fs i) ω) L (𝓝 (E.indicator (fun x => 1) ω))
aux : ∀ (ω : Ω), E.indicator (fun x => 1) ω = ↑(E.indicator (fun x => 1) ω)
⊢ μ E = ∫⁻ (ω : Ω), ↑(E.indicator (fun x => 1) ω) ∂μ

tactic: simp_rw [← aux, lintegral_indicator _ E_mble]
state_before: case h.e'_5.h.e'_3
Ω : Type u_1
inst✝⁶ : TopologicalSpace Ω
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝³ : L.IsCountablyGenerated
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ι → Ω →ᵇ ℝ≥0
fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => (fs i) ω) L (𝓝 (E.indicator (fun x => 1) ω))
aux : ∀ (ω : Ω), E.indicator (fun x => 1) ω = ↑(E.indicator (fun x => 1) ω)
⊢ μ E = ∫⁻ (ω : Ω), ↑(E.indicator (fun x => 1) ω) ∂μ
state_after: case h.e'_5.h.e'_3
Ω : Type u_1
inst✝⁶ : TopologicalSpace Ω
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝³ : L.IsCountablyGenerated
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ι → Ω →ᵇ ℝ≥0
fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => (fs i) ω) L (𝓝 (E.indicator (fun x => 1) ω))
aux : ∀ (ω : Ω), E.indicator (fun x => 1) ω = ↑(E.indicator (fun x => 1) ω)
⊢ μ E = ∫⁻ (x : Ω) in E, 1 ∂μ

tactic: simp only [lintegral_one, Measure.restrict_apply, MeasurableSet.univ, univ_inter]
state_before: case h.e'_5.h.e'_3
Ω : Type u_1
inst✝⁶ : TopologicalSpace Ω
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝³ : L.IsCountablyGenerated
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ι → Ω →ᵇ ℝ≥0
fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => (fs i) ω) L (𝓝 (E.indicator (fun x => 1) ω))
aux : ∀ (ω : Ω), E.indicator (fun x => 1) ω = ↑(E.indicator (fun x => 1) ω)
⊢ μ E = ∫⁻ (x : Ω) in E, 1 ∂μ
state_after: no goals

tactic: simp only [ENNReal.coe_indicator, ENNReal.coe_one]
state_before: Ω : Type u_1
inst✝⁶ : TopologicalSpace Ω
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝³ : L.IsCountablyGenerated
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ι → Ω →ᵇ ℝ≥0
fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => (fs i) ω) L (𝓝 (E.indicator (fun x => 1) ω))
ω : Ω
⊢ E.indicator (fun x => 1) ω = ↑(E.indicator (fun x => 1) ω)
state_after: no goals