= Uniform ergodic convergence for uniquely ergodic systems
For a <uniquely ergodic> continuous map $T$ of a compact <metric space>, every continuous $f$ satisfies $N^{-1}\sum_{j=0}^{N-1}f(T^jx)\to\int f\,d\mu$ uniformly in $x$. Otherwise choose increasingly long orbit <empirical measures> with a fixed discrepancy. Compactness provides a weak limit; the telescoping identity makes it invariant, so it must be the unique measure $\mu$, contradicting that discrepancy. Compactness and continuity are essential hypotheses.
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