ProbabilityTheory.measurable_kernel_iff
No docstring.
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𝓧 : Type u_1m𝓧 : MeasurableSpace 𝓧A measurable space is a space equipped with a σ-algebra. -
𝓨 : Type u_2m𝓨 : MeasurableSpace 𝓨 -
𝓩 : Type u_3m𝓩 : MeasurableSpace 𝓩
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f : 𝓧 → Kernel 𝓨 𝓩A kernel from a measurable spaceαto another measurable spaceβis a measurable functionκ : α → Measure β.
Measurable f ↔ ∀ (y : 𝓨), Measurable fun x => (f x) yMeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
ProbabilityTheory.Kernel : (α : Type u_1) → (β : Type u_2) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_1 u_2)A kernel from a measurable space `α` to another measurable space `β` is a measurable function `κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by `MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`.
Iff : Prop → Prop → PropIf and only if, or logical bi-implication. `a ↔ b` means that `a` implies `b` and vice versa. By `propext`, this implies that `a` and `b` are equal and hence any expression involving `a` is equivalent to the corresponding expression with `b` instead. Conventions for notations in identifiers: * The recommended spelling of `↔` in identifiers is `iff`. * The recommended spelling of `<->` in identifiers is `iff` (prefer `↔` over `<->`).
Measurable : {α : Type u_1} → {β : Type u_2} → [MeasurableSpace α] → [MeasurableSpace β] → (α → β) → PropA function `f` between measurable spaces is measurable if the preimage of every measurable set is measurable.
Code
lemma measurable_kernel_iff (f : 𝓧 → Kernel 𝓨 𝓩) : Measurable f ↔ ∀ y, Measurable (f · y)
Proof
by unfold instMeasurableSpaceKernel rw [measurable_comap_iff, measurable_pi_iff] simp
New in v4.34.0-rc2-78-g3ff93c0 (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
Audit surface: 1 project declarations, 11 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.