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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_{iThis page claims that this measure is in fact ergodic.

Question: Why is $\mu_{x}$ ergodic when $x$ is generic?

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$x$ 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.