/-- The **Lusin-Souslin theorem**: the range of a continuous injective function defined on a Polish
space is Borel-measurable. -/
theorem measurableSet_range_of_continuous_injective {β : Type*} [TopologicalSpace γ]
    [PolishSpace γ] [TopologicalSpace β] [T2Space β] [MeasurableSpace β] [OpensMeasurableSpace β]
    {f : γ → β} (f_cont : Continuous f) (f_inj : Injective f) :
    MeasurableSet (range f) := by
  /- We follow [Fremlin, *Measure Theory* (volume 4, 423I)][fremlin_vol4].
    Let `b = {s i}` be a countable basis for `α`. When `s i` and `s j` are disjoint, their images
    are disjoint analytic sets, hence by the separation theorem one can find a Borel-measurable set
    `q i j` separating them.
    Let `E i = closure (f '' s i) ∩ ⋂ j, q i j \ q j i`. It contains `f '' (s i)` and it is
    measurable. Let `F n = ⋃ E i`, where the union is taken over those `i` for which `diam (s i)`
    is bounded by some number `u n` tending to `0` with `n`.
    We claim that `range f = ⋂ F n`, from which the measurability is obvious. The inclusion `⊆` is
    straightforward. To show `⊇`, consider a point `x` in the intersection. For each `n`, it belongs
    to some `E i` with `diam (s i) ≤ u n`. Pick a point `y i ∈ s i`. We claim that for such `i`
    and `j`, the intersection `s i ∩ s j` is nonempty: if it were empty, then thanks to the
    separating set `q i j` in the definition of `E i` one could not have `x ∈ E i ∩ E j`.
    Since these two sets have small diameter, it follows that `y i` and `y j` are close.
    Thus, `y` is a Cauchy sequence, converging to a limit `z`. We claim that `f z = x`, completing
    the proof.
    Otherwise, one could find open sets `v` and `w` separating `f z` from `x`. Then, for large `n`,
    the image `f '' (s i)` would be included in `v` by continuity of `f`, so its closure would be
    contained in the closure of `v`, and therefore it would be disjoint from `w`. This is a
    contradiction since `x` belongs both to this closure and to `w`. -/
  letI := upgradePolishSpace γ
  obtain ⟨b, b_count, b_nonempty, hb⟩ :
    ∃ b : Set (Set γ), b.Countable ∧ ∅ ∉ b ∧ IsTopologicalBasis b := exists_countable_basis γ
  haveI : Encodable b := b_count.toEncodable
  let A := { p : b × b // Disjoint (p.1 : Set γ) p.2 }
  -- for each pair of disjoint sets in the topological basis `b`, consider Borel sets separating
  -- their images, by injectivity of `f` and the Lusin separation theorem.
  have : ∀ p : A, ∃ q : Set β,
      f '' (p.1.1 : Set γ) ⊆ q ∧ Disjoint (f '' (p.1.2 : Set γ)) q ∧ MeasurableSet q := by
    intro p
    apply
      AnalyticSet.measurablySeparable ((hb.isOpen p.1.1.2).analyticSet_image f_cont)
        ((hb.isOpen p.1.2.2).analyticSet_image f_cont)
    

Proof state:
α : Type u_1
ι : Type u_2
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous f
f_inj : Injective f
this✝ : UpgradedPolishSpace γ := upgradePolishSpace γ
b : Set (Set γ)
b_count : Set.Countable b
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this : Encodable ↑b
A : Type u_3 := { p // Disjoint ↑p.1 ↑p.2 }
p : A
⊢ Disjoint (f '' ↑(↑p).1) (f '' ↑(↑p).2)

Next tactic:
exact Disjoint.image p.2 (f_inj.injOn univ) (subset_univ _) (subset_univ _)