ProbabilityTheory.Kernel.instIsMarkovKernelSigmaSigma
No docstring.
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α : Type u_1 -
γ : Type u_2mγ : MeasurableSpace γA measurable space is a space equipped with a σ-algebra.
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β : α → Type u_3(a : α) → MeasurableSpace (β a) -
κ : (a : α) → Kernel (β a) γA kernel from a measurable spaceαto another measurable spaceβis a measurable functionκ : α → Measure β.∀ (a : α), IsMarkovKernel (κ a)A kernel is a Markov kernel if every measure in its image is a probability measure.
IsMarkovKernel (sigma κ)MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
ProbabilityTheory.IsMarkovKernel : {α : Type u_1} →
{β : Type u_2} → {mα : MeasurableSpace α} → {mβ : MeasurableSpace β} → ProbabilityTheory.Kernel α β → PropA kernel is a Markov kernel if every measure in its image is a probability measure.
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)`.
ProbabilityTheory.Kernel.sigma : {α : Type u_1} →
{γ : Type u_2} →
{β : α → Type u_3} →
[inst : (a : α) → MeasurableSpace (β a)] →
{mγ : MeasurableSpace γ} →
((a : α) → ProbabilityTheory.Kernel (β a) γ) → ProbabilityTheory.Kernel ((a : α) × β a) γThe kernel on `Σ a, β a` which is `κ a` on the fiber `β a`.Go to its page
Code
instance (κ : (a : α) → Kernel (β a) γ) [∀ a, IsMarkovKernel (κ a)] : IsMarkovKernel (sigma κ)
Proof
⟨fun x ↦ (IsMarkovKernel.isProbabilityMeasure (κ := κ x.1) x.2)⟩
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
Audit surface: 1 project declarations, 13 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.