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feat: a measurable cylinder is measurable (#16939)
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Add this easy lemma.
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Etienne committed Sep 20, 2024
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6 changes: 6 additions & 0 deletions Mathlib/MeasureTheory/Constructions/Cylinders.lean
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Expand Up @@ -272,6 +272,12 @@ theorem mem_measurableCylinders (t : Set (∀ i, α i)) :
t ∈ measurableCylinders α ↔ ∃ s S, MeasurableSet S ∧ t = cylinder s S := by
simp_rw [measurableCylinders, mem_iUnion, exists_prop, mem_singleton_iff]

@[measurability]
theorem _root_.MeasurableSet.of_mem_measurableCylinders {s : Set (Π i, α i)}
(hs : s ∈ measurableCylinders α) : MeasurableSet s := by
obtain ⟨I, t, mt, rfl⟩ := (mem_measurableCylinders s).1 hs
exact mt.cylinder

/-- A finset `s` such that `t = cylinder s S`. `S` is given by `measurableCylinders.set`. -/
noncomputable def measurableCylinders.finset (ht : t ∈ measurableCylinders α) : Finset ι :=
((mem_measurableCylinders t).mp ht).choose
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