= Compactness of probability measures on a compact metric space
On a <compact metric space>, every sequence of <Borel probability measures> has a subsequence with <weak convergence of probability measures> to a <Borel probability measure>. To see this, choose a countable uniformly dense subset of continuous functions, extract a diagonal subsequence of their bounded integrals, and extend the resulting positive normalized functional to all continuous functions. The <Riesz representation theorem> supplies its <Borel probability measure>. This is useful for constructing an <invariant measure> from orbit <empirical measures>.
Back to article page
