= Krylov–Bogolyubov theorem
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= Krylov-Bogolyubov time-average argument
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For a <Feller semigroup> on a compact <metric space>, time-averaged transition laws have weakly convergent subsequences by <compactness of probability measures on a compact metric space>. Shifting a time interval of length $n$ by $t$ changes a bounded test-function average by at most $2t\|f\|_\infty/n$. A weak subsequential limit is therefore an <invariant probability measure for a semigroup>. More generally, tightness of the time averages supplies the required compactness.
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