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 by changes a bounded test-function average by at most . 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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The Krylov–Bogolyubov theorem, often associated with the works of Nikolai Krylov and Nikolai Bogolyubov, is a result in the theory of dynamical systems and statistical mechanics. It addresses the existence of invariant measures for certain classes of dynamical systems, particularly in the context of Hamiltonian systems and stochastic processes. In more technical terms, the theorem typically applies to systems that can be described by a flow in a finite-dimensional phase space.