Krylov–Bogolyubov existence theorem

Statement

Every continuous self-map $T:X\to X$ of a nonempty compact metric space admits an invariant Borel probability measure. This is the compact-space Krylov–Bogolyubov existence theorem.

Empirical-measure proof

Choose $x\in X$ and put $\mu_N=N^{-1}\sum_{j=0}^{N-1}\delta_{T^jx}$ . Probabilities on a compact metric space have a weakly convergent subsequence: convergence here means convergence of the integral of every continuous function. For any such subsequential limit $\mu$ and $f\in C(X)$ ,

\int(f\circ T-f)\,d\mu_N=\frac{f(T^Nx)-f(x)}N\longrightarrow0.

Continuity permits passage to the limit, giving $\int f\circ T\,d\mu=\int f\,d\mu$ , hence invariance. The theorem establishes existence; it does not assert uniqueness or convergence of the entire empirical-measure sequence.