Let $\mu$ be a limit point for the sequence of measures $$\frac{1}{n}\sum_{k=0}^{n-1}T_*^k\delta_x$$ for some point $x\in X$ and let the map $T_∗: \mathcal{M}(X)\rightarrow \mathcal{M}(X)$ be continuous at the measure $\mu$. Then $T_∗\mu = \mu$.
Above is a theorem. I want to ask, if we drop the condition that map is continuous, how to construct a counterexample?