Uniform averages characterize unique ergodicity

Statement

For a continuous $T:X\to X$ on a nonempty compact metric space, the following are equivalent:

There is exactly one invariant Borel probability $\mu$ .
For every $f\in C(X)$ , the averages $N^{-1}\sum_{n=0}^{N-1}f(T^nx)$ converge uniformly in $x$ to a constant.

The constant is $\int f\,d\mu$ . This is the uniform-averaging characterization of unique ergodicity.

Proof idea

If uniform convergence fails under uniqueness, choose starting points and lengths witnessing failure. A weakly convergent subsequence of their empirical measures has invariant limit, by the telescoping boundary term. Uniqueness forces that limit to be $\mu$ , contradicting the failed averages. Conversely, integrate uniform limits against any invariant probability; all such measures agree on every continuous function and are equal.

Scope

Minimality is not required. The theorem uses continuous observables and gives convergence for every starting point; Birkhoff's theorem permits integrable measurable observables but only gives almost-everywhere convergence.