Invariant measure ($T_{\ast }\mu =\mu$)

For a measurable transformation $T$, a measure is invariant if the pushforward measure equals the original measure, or equivalently $\mu (T^{-1}A)=\mu (A)$ for every measurable $A$. This is the measure used in a measure-preserving system.

A Borel probability measure is invariant for a Markov kernel semigroup if every transition preserves it. For a Feller semigroup on a compact metric space, it suffices to test the displayed identity on continuous functions. Kernel Jensen inequality gives contraction on $L^p(\mu )$, and sup-norm strong continuity followed by density gives strong continuity there for $1\le p<\infty$.

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 $2t\Vert f{\Vert }_{\infty }/n$. A weak subsequential limit is therefore an invariant probability measure for a semigroup. More generally, tightness of the time averages supplies the required compactness.