Plain-English statement
A Markov operator averages an observable against the conditional transition law from the current state.
Mathematical statement
P_t f(x) = integral f(y) K_t(x,dy).
Intuition
The transition kernel is the conditional law of the future state, so integration is conditional expectation.
Conditions
- a measurable state space
- Markov transition kernels
- a measurable ENNReal observable
Why these conditions cannot be dropped
- kernel integration needs measurability
- ENNReal permits nonnegative unbounded observables without hidden integrability
Proof route
- lintegrate the observable against K_t(x,dy)
- use lintegral_kernel to retain measurability in x
Lean interface notes
- the result is a subtype of measurable observables
- real bounded-continuous operators are constructed later by the Feller bridge
Read the mathematics first, then descend into Lean
You do not need to know Lean before opening this panel. The page keeps the paper-level theorem, rigorous proof obligations, exact Lean declaration, and proof dependencies as separate layers so a first-time reader can move down one layer at a time.
Proof architecture
Start with the tree when learning: prerequisites sit below the theorem and downstream results sit above it. Switch to the network when you want to understand where this declaration lives in the local formal library. Every mapped node is clickable.
Graph rule: only dependencies found by the ASTIS source scan are drawn. Missing tactic indirection is treated as an under-approximation; the site never invents an edge just to make a prettier graph.
How to read the exact Lean declaration
Read a Lean theorem left-to-right exactly as you would unpack a mathematical sentence: name → ambient types → automatically inferred structures → explicit hypotheses → conclusion → proof. Then read the proof top-to-bottom as transformations of the current goal.
- NameWhat reusable mathematical fact is being created?
- ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
- PropositionAfter the colon, translate the Lean expression back into a paper statement.
- Proof actionsAfter
by, ask what each tactic does to the mathematical goal—not only what syntax it uses.
Syntax used on this page
This glossary is filtered to syntax that actually occurs in the declaration above. Open a symbol only when you meet it, rather than memorizing Lean grammar in advance.
Source voice and ASTIS voice stay separate
When Samplinglib shows a short quotation from Chewi, it is labeled as a source excerpt and linked to the canonical book page. Intuition, expanded proof steps, hidden regularity assumptions, and Lean explanations are ASTIS-authored commentary. A quotation never substitutes for a formal proof, and an ASTIS explanation is never attributed to the textbook author.
Lean statement
def markovOperator {K : ℝ≥0 → Kernel E E}
(_hK : TransitionKernelContract K) (t : ℝ≥0) :
MeasurableENNReal E → MeasurableENNReal E :=
fun f => by
exact ⟨fun x => ∫⁻ y, f y ∂K t x, f.2.lintegral_kernel⟩
/-- A Markov operator preserves constant nonnegative observables. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovSemigroup.lean:54published source at 7bcd37294df1