/-- `liminf` over a general filter is measurable. See `measurable_liminf` for the version over `ℕ`.
-/
theorem measurable_liminf' {ι ι'} {f : ι → δ → α} {v : Filter ι} (hf : ∀ i, Measurable (f i))
    {p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasCountableBasis p s) (hs : ∀ j, (s j).Countable) :
    Measurable fun x => liminf (fun i => f i x) v := by
  /- We would like to write the liminf as `⨆ (j : Subtype p), ⨅ (i : s j), f i x`, as the
  measurability would follow from the measurability of infs and sups. Unfortunately, this is not
  true in general conditionally complete linear orders because of issues with empty sets or sets
  which are not bounded above or below. A slightly more complicated expression for the liminf,
  valid in general, is given in `Filter.HasBasis.liminf_eq_ite`. This expression, built from
  `if ... then ... else` and infs and sups, can be readily checked to be measurable. -/
  have : Countable (Subtype p) := Encodable.nonempty_encodable.1 hv.countable
  rcases isEmpty_or_nonempty (Subtype p) with hp|hp
  · simp [hv.liminf_eq_sSup_iUnion_iInter]
  by_cases H : ∃ (j : Subtype p), s j = ∅
  · simp_rw [hv.liminf_eq_ite, if_pos H, measurable_const]
  simp_rw [hv.liminf_eq_ite, if_neg H]
  have : ∀ i, Countable (s i) := fun i ↦ countable_coe_iff.2 (hs i)
  let m : Subtype p → Set δ := fun j ↦ {x | BddBelow (range (fun (i : s j) ↦ f i x))}
  have m_meas : ∀ j, MeasurableSet (m j) :=
    fun j ↦ measurableSet_bddBelow_range (fun (i : s j) ↦ hf i)
  have mc_meas : MeasurableSet {x | ∀ (j : Subtype p), x ∉ m j} := by
    rw [setOf_forall]
    exact MeasurableSet.iInter (fun j ↦ (m_meas j).compl)
  apply Measurable.piecewise mc_meas measurable_const
  apply measurable_iSup (fun j ↦ ?_)
  let reparam : δ → Subtype p → Subtype p := fun x ↦ liminf_reparam (fun i ↦ f i x) s p
  let F0 : Subtype p → δ → α := fun j x ↦ ⨅ (i : s j), f i x
  have F0_meas : ∀ j, Measurable (F0 j) := fun j ↦ measurable_iInf (fun (i : s j) ↦ hf i)
  set F1 : δ → α := fun x ↦ F0 (reparam x j) x with hF1
  change Measurable F1
  let g : ℕ → Subtype p := choose (exists_surjective_nat (Subtype p))
  have Z : ∀ x, ∃ n, x ∈ m (g n) ∨ ∀ k, x ∉ m k := by
    intro x
    by_cases H : ∃ k, x ∈ m k
    · rcases H with ⟨k, hk⟩
      rcases choose_spec (exists_surjective_nat (Subtype p)) k with ⟨n, rfl⟩
      exact ⟨n, Or.inl hk⟩
    · push_neg at H
      exact ⟨0, Or.inr H⟩
  have : F1 = fun x ↦ if x ∈ m j then F0 j x else F0 (g (Nat.find (Z x))) x := by
    ext x
    have A : reparam x j = if x ∈ m j then j else g (Nat.find (Z x)) := rfl
    split_ifs with hjx
    · have : reparam x j = j := by rw [A, if_pos hjx]
      simp only [hF1, this]
    · have : reparam x j = g (Nat.find (Z x)) := by rw [A, if_neg hjx]
      simp only [hF1, this]
  rw [this]
  apply Measurable.piecewise (m_meas j) (F0_meas j)
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
ι' : Type u_7
f : ι → δ → α
v : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hv : HasCountableBasis v p s
hs : ∀ (j : ι'), Set.Countable (s j)
this✝¹ : Countable (Subtype p)
hp : Nonempty (Subtype p)
H : ¬∃ j, s ↑j = ∅
this✝ : ∀ (i : ι'), Countable ↑(s i)
m : Subtype p → Set δ := fun j => {x | BddBelow (range fun i => f (↑i) x)}
m_meas : ∀ (j : Subtype p), MeasurableSet (m j)
mc_meas : MeasurableSet {x | ∀ (j : Subtype p), x ∉ m j}
j : Subtype p
reparam : δ → Subtype p → Subtype p := fun x => liminf_reparam (fun i => f i x) s p
F0 : Subtype p → δ → α := fun j x => ⨅ i, f (↑i) x
F0_meas : ∀ (j : Subtype p), Measurable (F0 j)
F1 : δ → α := fun x => F0 (reparam x j) x
hF1 : F1 = fun x => F0 (reparam x j) x
g : ℕ → Subtype p := choose (_ : ∃ f, Function.Surjective f)
Z : ∀ (x : δ), ∃ n, x ∈ m (g n) ∨ ∀ (k : Subtype p), x ∉ m k
this : F1 = fun x => if x ∈ m j then F0 j x else F0 (g (Nat.find (_ : ∃ n, x ∈ m (g n) ∨ ∀ (k : Subtype p), x ∉ m k))) x
⊢ Measurable fun x => F0 (g (Nat.find (_ : ∃ n, x ∈ m (g n) ∨ ∀ (k : Subtype p), x ∉ m k))) x

Next tactic:
apply Measurable.find (fun n ↦ F0_meas (g n)) (fun n ↦ ?_)