= Generic point for an invariant measure
{title2=$\frac1n\sum_{k=0}^{n-1}\varphi(f^kx)\longrightarrow\int\varphi\,d\mu$}
= Generic points for an invariant measure
{synonym}
A point is generic for an <invariant measure> $\mu$ of a <continuous map> on a <compact metric space> if the displayed convergence holds for every <continuous function> $\varphi$. Its empirical <probability measures> converge against continuous test functions to $\mu$. The same test-function condition may be used before invariance is known: the telescoping identity for $\varphi\circ f-\varphi$ then proves that the limiting <Borel probability measure> must be invariant. If almost every point is generic for the same measure, continuous approximation of invariant-set indicators proves that this measure is <ergodic>.
Back to article page
