= Invariant probability measure for a semigroup
{title2=$\int P_tf\,d\mu=\int f\,d\mu$}
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 $1\leq p<\infty$.
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