measurableEmbedding_prodMk_left_of_measurableSet
From the authors
Prod.mk x is a measurable embedding as soon as {x} is measurable. This generalises
measurableEmbedding_prodMk_left, which assumes MeasurableSingletonClass.
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α : Type u_1MeasurableSpace αA measurable space is a space equipped with a σ-algebra. -
β : Type u_2MeasurableSpace β
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x : α
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hx : MeasurableSet {x}MeasurableSet smeans thatsis measurable (in the ambient measure space onα)
MeasurableEmbedding (Prod.mk x)f : α → β is called a *measurable embedding* if it is injective, measurable, and sends measurable sets to measurable sets.MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
MeasurableSet : {α : Type u_1} → [MeasurableSpace α] → Set α → Prop`MeasurableSet s` means that `s` is measurable (in the ambient measure space on `α`)
Singleton.singleton : {α : outParam (Type u)} → {β : Type v} → [self : Singleton α β] → α → β`singleton x` is a collection with the single element `x` (notation: `{x}`).
Conventions for notations in identifiers:
* The recommended spelling of `{x}` in identifiers is `singleton`.MeasurableEmbedding : {α : Type u_1} → {β : Type u_2} → [MeasurableSpace α] → [MeasurableSpace β] → (α → β) → PropA map `f : α → β` is called a *measurable embedding* if it is injective, measurable, and sends measurable sets to measurable sets. The latter assumption can be replaced with “`f` has measurable inverse `g : Set.range f → α`”, see `MeasurableEmbedding.measurable_rangeSplitting`, `MeasurableEmbedding.of_measurable_inverse_range`, and `MeasurableEmbedding.of_measurable_inverse`. One more interpretation: `f` is a measurable embedding if it defines a measurable equivalence to its range and the range is a measurable set. One implication is formalized as `MeasurableEmbedding.equivRange`; the other one follows from `MeasurableEquiv.measurableEmbedding`, `MeasurableEmbedding.subtype_coe`, and `MeasurableEmbedding.comp`.
Prod.mk : {α : Type u} → {β : Type v} → α → β → α × βConstructs a pair. This is usually written `(x, y)` instead of `Prod.mk x y`. Conventions for notations in identifiers: * The recommended spelling of `(a, b)` in identifiers is `mk`.
Code
lemma measurableEmbedding_prodMk_left_of_measurableSet {α β : Type*} [MeasurableSpace α]
[MeasurableSpace β] {x : α} (hx : MeasurableSet {x}) :
MeasurableEmbedding (Prod.mk x : β → α × β) where
injective _ _ hProof
(Prod.ext_iff.mp h).2
measurable := by fun_prop
measurableSet_image' s hs := by
convert! hx.prod hs
ext p
simp [Prod.ext_iff, eq_comm, and_left_comm]New in v4.34.0-rc2-74-ge05e4f3 (2026-09-10), 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, 9 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.