Let $X=\{0,1\}^{\mathbb{N}}$, $T:X\to X$ the shift map, and $\mu$ a $T$-invariant probability measure on $X$. A point $x \in X$ is generic if $ \lim\, \frac{1}{n}\sum_{i
Question: Why is $\mu_{x}$ ergodic when $x$ is generic?
Let $X=\{0,1\}^{\mathbb{N}}$, $T:X\to X$ the shift map, and $\mu$ a $T$-invariant probability measure on $X$. A point $x \in X$ is generic if $ \lim\, \frac{1}{n}\sum_{i
Question: Why is $\mu_{x}$ ergodic when $x$ is generic?
$x$ can be generic by your definition while $\mu_x$ is not ergodic:
A generic point for a non-ergodic measure
The page you linked is only considering points that are generic for ergodic measures.