Theorem: MeasureTheory.FiniteMeasure.limsup_measure_closed_le_of_tendsto
File: Mathlib/MeasureTheory/Measure/Portmanteau.lean

Proof steps:
state_before: Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F
state_after: case inl
Ω : Type u_1
ι : Type u_2
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
F : Set Ω
F_closed : IsClosed F
μs_lim : Tendsto μs ⊥ (𝓝 μ)
⊢ limsup (fun i => ↑(μs i) F) ⊥ ≤ ↑μ F

case inr
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F
tactic: rcases L.eq_or_neBot with rfl | hne
state_before: case inr
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F
state_after: case inr.h
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
⊢ ∀ (ε : ℝ≥0), 0 < ε → ↑μ F < ⊤ → limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
tactic: apply ENNReal.le_of_forall_pos_le_add
state_before: case inr.h
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
⊢ ∀ (ε : ℝ≥0), 0 < ε → ↑μ F < ⊤ → limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
state_after: case inr.h
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
tactic: intro ε ε_pos _
state_before: case inr.h
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
state_after: case inr.h
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
tactic: let fs := F_closed.apprSeq
state_before: case inr.h
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
state_after: case inr.h
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
tactic: have key₁ : Tendsto (fun n ↦ ∫⁻  ω, (fs n ω : ℝ≥0∞) ∂μ) atTop (𝓝 ((μ : Measure Ω) F)) :=
    HasOuterApproxClosed.tendsto_lintegral_apprSeq F_closed (μ : Measure Ω)
state_before: case inr.h
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
state_after: case inr.h
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
tactic: have room₁ : (μ : Measure Ω) F < (μ : Measure Ω) F + ε / 2 := by
    apply
      ENNReal.lt_add_right (measure_lt_top (μ : Measure Ω) F).ne
        (ENNReal.div_pos_iff.mpr ⟨(ENNReal.coe_pos.mpr ε_pos).ne.symm, ENNReal.two_ne_top⟩).ne.symm
state_before: case inr.h
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
state_after: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
tactic: rcases eventually_atTop.mp (eventually_lt_of_tendsto_lt room₁ key₁) with ⟨M, hM⟩
state_before: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
state_after: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
key₂ : Tendsto (fun i => ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs i)) L (𝓝 (∫⁻ (x : Ω), ↑((fs M) x) ∂↑μ))
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
tactic: have key₂ := FiniteMeasure.tendsto_iff_forall_lintegral_tendsto.mp μs_lim (fs M)
state_before: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
key₂ : Tendsto (fun i => ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs i)) L (𝓝 (∫⁻ (x : Ω), ↑((fs M) x) ∂↑μ))
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
state_after: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
key₂ : Tendsto (fun i => ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs i)) L (𝓝 (∫⁻ (x : Ω), ↑((fs M) x) ∂↑μ))
room₂ : ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ < ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
tactic: have room₂ :
    (lintegral (μ : Measure Ω) fun a => fs M a) <
      (lintegral (μ : Measure Ω) fun a => fs M a) + ε / 2 := by
    apply ENNReal.lt_add_right (ne_of_lt ?_)
        (ENNReal.div_pos_iff.mpr ⟨(ENNReal.coe_pos.mpr ε_pos).ne.symm, ENNReal.two_ne_top⟩).ne.symm
    apply BoundedContinuousFunction.lintegral_lt_top_of_nnreal
state_before: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
key₂ : Tendsto (fun i => ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs i)) L (𝓝 (∫⁻ (x : Ω), ↑((fs M) x) ∂↑μ))
room₂ : ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ < ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
state_after: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
key₂ : Tendsto (fun i => ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs i)) L (𝓝 (∫⁻ (x : Ω), ↑((fs M) x) ∂↑μ))
room₂ : ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ < ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near : ∀ᶠ (x : ι) in L, ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs x) ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
tactic: have ev_near := Eventually.mono (eventually_lt_of_tendsto_lt room₂ key₂) fun n => le_of_lt
state_before: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
key₂ : Tendsto (fun i => ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs i)) L (𝓝 (∫⁻ (x : Ω), ↑((fs M) x) ∂↑μ))
room₂ : ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ < ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near : ∀ᶠ (x : ι) in L, ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs x) ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
state_after: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
key₂ : Tendsto (fun i => ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs i)) L (𝓝 (∫⁻ (x : Ω), ↑((fs M) x) ∂↑μ))
room₂ : ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ < ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near : ∀ᶠ (x : ι) in L, ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs x) ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near' : ∀ᶠ (x : ι) in L, ↑(μs x) F ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
tactic: have ev_near' := Eventually.mono ev_near
    (fun n ↦ le_trans (HasOuterApproxClosed.measure_le_lintegral F_closed (μs n) M))
state_before: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
key₂ : Tendsto (fun i => ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs i)) L (𝓝 (∫⁻ (x : Ω), ↑((fs M) x) ∂↑μ))
room₂ : ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ < ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near : ∀ᶠ (x : ι) in L, ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs x) ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near' : ∀ᶠ (x : ι) in L, ↑(μs x) F ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
⊢ limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε
state_after: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
key₂ : Tendsto (fun i => ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs i)) L (𝓝 (∫⁻ (x : Ω), ↑((fs M) x) ∂↑μ))
room₂ : ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ < ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near : ∀ᶠ (x : ι) in L, ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs x) ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near' : ∀ᶠ (x : ι) in L, ↑(μs x) F ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
⊢ limsup (fun x => ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2) L ≤ ↑μ F + ↑ε
tactic: apply (Filter.limsup_le_limsup ev_near').trans
state_before: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
key₂ : Tendsto (fun i => ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs i)) L (𝓝 (∫⁻ (x : Ω), ↑((fs M) x) ∂↑μ))
room₂ : ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ < ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near : ∀ᶠ (x : ι) in L, ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs x) ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near' : ∀ᶠ (x : ι) in L, ↑(μs x) F ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
⊢ limsup (fun x => ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2) L ≤ ↑μ F + ↑ε
state_after: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
key₂ : Tendsto (fun i => ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs i)) L (𝓝 (∫⁻ (x : Ω), ↑((fs M) x) ∂↑μ))
room₂ : ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ < ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near : ∀ᶠ (x : ι) in L, ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs x) ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near' : ∀ᶠ (x : ι) in L, ↑(μs x) F ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
⊢ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2 ≤ ↑μ F + ↑ε
tactic: rw [limsup_const]
state_before: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
key₂ : Tendsto (fun i => ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs i)) L (𝓝 (∫⁻ (x : Ω), ↑((fs M) x) ∂↑μ))
room₂ : ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ < ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near : ∀ᶠ (x : ι) in L, ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs x) ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near' : ∀ᶠ (x : ι) in L, ↑(μs x) F ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
⊢ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2 ≤ ↑μ F + ↑ε
state_after: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
key₂ : Tendsto (fun i => ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs i)) L (𝓝 (∫⁻ (x : Ω), ↑((fs M) x) ∂↑μ))
room₂ : ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ < ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near : ∀ᶠ (x : ι) in L, ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs x) ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near' : ∀ᶠ (x : ι) in L, ↑(μs x) F ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
⊢ ↑μ F + ↑ε / 2 + ↑ε / 2 ≤ ↑μ F + ↑ε
tactic: apply le_trans (add_le_add (hM M rfl.le).le (le_refl (ε / 2 : ℝ≥0∞)))
state_before: case inr.h.intro
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F < ⊤
fs : ℕ → Ω →ᵇ ℝ≥0 := F_closed.apprSeq
key₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂↑μ) atTop (𝓝 (↑μ F))
room₁ : ↑μ F < ↑μ F + ↑ε / 2
M : ℕ
hM : ∀ b ≥ M, ∫⁻ (ω : Ω), ↑((fs b) ω) ∂↑μ < ↑μ F + ↑ε / 2
key₂ : Tendsto (fun i => ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs i)) L (𝓝 (∫⁻ (x : Ω), ↑((fs M) x) ∂↑μ))
room₂ : ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ < ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near : ∀ᶠ (x : ι) in L, ∫⁻ (x : Ω), ↑((fs M) x) ∂↑(μs x) ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
ev_near' : ∀ᶠ (x : ι) in L, ↑(μs x) F ≤ ∫⁻ (a : Ω), ↑((fs M) a) ∂↑μ + ↑ε / 2
⊢ ↑μ F + ↑ε / 2 + ↑ε / 2 ≤ ↑μ F + ↑ε
state_after: no goals
tactic: simp only [add_assoc, ENNReal.add_halves, le_refl]