`Measurable.sigmaMk`🔗

Lemma

From the authors

x ↦ ⟨n x, f (n x) x⟩ is measurable when the index n x ranges over a countable type with measurable singletons and each f i is measurable.

Types

α : Type u_1Countable αA type α is countable if there exists an injective map α → ℕ.MeasurableSpace αA measurable space is a space equipped with a σ-algebra.MeasurableSingletonClass αA typeclass mixin for MeasurableSpaces such that each singleton is measurable.
γ : Type u_2MeasurableSpace γ

Given

β : α → Type u_3(a : α) → MeasurableSpace (β a)
n : γ → α
f : (a : α) → γ → β a

Assuming

hn : Measurable nA function f between measurable spaces is measurable if the preimage of every measurable set is measurable.
hf : ∀ (a : α), Measurable (f a)

Then

Measurable fun x => ⟨n x, f (n x) x⟩

Code

lemma Measurable.sigmaMk [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
    {n : γ → α} (hn : Measurable n) {f : (a : α) → γ → β a} (hf : ∀ a, Measurable (f a)) :
    Measurable fun x ↦ (⟨n x, f (n x) x⟩ : Σ a, β a)

Proof

by
  intro s hs
  have : (fun x ↦ (⟨n x, f (n x) x⟩ : Σ a, β a)) ⁻¹' s =
      ⋃ a, n ⁻¹' {a} ∩ f a ⁻¹' (Sigma.mk a ⁻¹' s) := by
    ext x
    simp only [Set.mem_preimage, Set.mem_iUnion, Set.mem_inter_iff, Set.mem_singleton_iff]
    constructor
    · intro h
      exact ⟨n x, rfl, h⟩
    · rintro ⟨a, rfl, h⟩
      exact h
  rw [this]
  exact MeasurableSet.iUnion fun a ↦
    (hn (measurableSet_singleton a)).inter (hf a (measurableSet_sigma_iff.1 hs a))

New in v4.34.0-rc2-82-ga6c27a7 (2026-09-13), and its meaning has not changed since.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Nothing to draw. Its statement rests on no other declaration in this project, and names nothing from a package left unaudited — so the graph is this declaration alone. That is the answer, not a missing picture.

Audit surface: 0 project declarations, 7 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.

Measurable.sigmaMk🔗

`Measurable.sigmaMk`🔗