I wonder if we could put "Almost Continuous" instead of "Continuous" in the theorem 1.7 of Peter Walters :
Let $X$ be a compact metric space, $\mathcal{B}(X)$ the $\sigma$-algebra of Borel subsets of $X$ and let $m$ be a probability measure on $(X,\mathcal{B}(X))$ such that $m(U) > 0$ for every non-empty open sets $U$. Suppose $T:X \to X$ is a continuous transformation which preserves the measure $m$ and is ergodic. Then almost all points of $X$ have a dense orbit under $T$ i.e. $\{x \in X | (T^{n}x)_{n=0}^{\infty} \text{is a dense subset of X} \}$ has $m$-measure 1.
Could we have result for almost continuous transformations?!
And Generally ergodicity means almost all points of the space are minimal? (If so doesn't matter $T$ is continuous or not?)
Thank you for helping me understand.