/-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup_measure_closed_le_of_forall_tendsto_measure
    {Ω ι : Type*} {L : Filter ι} [NeBot L]
    [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω]
    {μ : Measure Ω} [IsFiniteMeasure μ] {μs : ι → Measure Ω}
    (h : ∀ {E : Set Ω}, MeasurableSet E → μ (frontier E) = 0 →
            Tendsto (fun i ↦ μs i E) L (𝓝 (μ E)))
    (F : Set Ω) (F_closed : IsClosed F) :
    L.limsup (fun i ↦ μs i F) ≤ μ F := by
  have ex := exists_null_frontiers_thickening μ F
  let rs := Classical.choose ex
  have rs_lim : Tendsto rs atTop (𝓝 0) := (Classical.choose_spec ex).1
  have rs_pos : ∀ n, 0 < rs n := fun n ↦ ((Classical.choose_spec ex).2 n).1
  have rs_null : ∀ n, μ (frontier (Metric.thickening (rs n) F)) = 0 :=
    fun n ↦ ((Classical.choose_spec ex).2 n).2
  have Fthicks_open : ∀ n, IsOpen (Metric.thickening (rs n) F) :=
    fun n ↦ Metric.isOpen_thickening
  have key := fun (n : ℕ) ↦ h (Fthicks_open n).measurableSet (rs_null n)
  apply ENNReal.le_of_forall_pos_le_add
  intros ε ε_pos μF_finite
  have keyB := tendsto_measure_cthickening_of_isClosed (μ := μ) (s := F)
                ⟨1, ⟨by simp only [gt_iff_lt, zero_lt_one], measure_ne_top _ _⟩⟩ F_closed
  have nhd : Iio (μ F + ε) ∈ 𝓝 (μ F) := by
    apply Iio_mem_nhds
    exact ENNReal.lt_add_right μF_finite.ne (ENNReal.coe_pos.mpr ε_pos).ne'
  specialize rs_lim (keyB nhd)
  simp only [mem_map, mem_atTop_sets, ge_iff_le, mem_preimage, mem_Iio] at rs_lim
  obtain ⟨m, hm⟩ := rs_lim
  have aux' := fun i ↦ measure_mono (μ := μs i) (Metric.self_subset_thickening (rs_pos m) F)
  have aux : (fun i ↦ (μs i F)) ≤ᶠ[L] (fun i ↦ μs i (Metric.thickening (rs m) F)) :=
    eventually_of_forall aux'
  refine (limsup_le_limsup aux).trans ?_
  rw [Tendsto.limsup_eq (key m)]
  

Proof state:
case h.intro
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h : ∀ {E : Set Ω}, MeasurableSet E → ↑↑μ (frontier E) = 0 → Tendsto (fun i => ↑↑(μs i) E) L (𝓝 (↑↑μ E))
F : Set Ω
F_closed : IsClosed F
ex : ∃ rs, Tendsto rs atTop (𝓝 0) ∧ ∀ (n : ℕ), 0 < rs n ∧ ↑↑μ (frontier (Metric.thickening (rs n) F)) = 0
rs : ℕ → ℝ := Classical.choose ex
rs_pos : ∀ (n : ℕ), 0 < rs n
rs_null : ∀ (n : ℕ), ↑↑μ (frontier (Metric.thickening (rs n) F)) = 0
Fthicks_open : ∀ (n : ℕ), IsOpen (Metric.thickening (rs n) F)
key : ∀ (n : ℕ), Tendsto (fun i => ↑↑(μs i) (Metric.thickening (rs n) F)) L (𝓝 (↑↑μ (Metric.thickening (rs n) F)))
ε : ℝ≥0
ε_pos : 0 < ε
μF_finite : ↑↑μ F < ⊤
keyB : Tendsto (fun r => ↑↑μ (Metric.cthickening r F)) (𝓝 0) (𝓝 (↑↑μ F))
nhd : Iio (↑↑μ F + ↑ε) ∈ 𝓝 (↑↑μ F)
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → ↑↑μ (Metric.cthickening (Classical.choose ex b) F) < ↑↑μ F + ↑ε
aux' : ∀ (i : ι), ↑↑(μs i) F ≤ ↑↑(μs i) (Metric.thickening (rs m) F)
aux : (fun i => ↑↑(μs i) F) ≤ᶠ[L] fun i => ↑↑(μs i) (Metric.thickening (rs m) F)
⊢ ↑↑μ (Metric.thickening (rs m) F) ≤ ↑↑μ F + ↑ε

Next tactic:
apply (measure_mono (Metric.thickening_subset_cthickening (rs m) F)).trans (hm m rfl.le).le