# Ergodicity with respect to the shift

On the space $S=\{ 0,1,\ldots,m \}^{\mathbb{N}}$ for some $m\in \mathbb{Z}_{+}.$ And given a probability $\mu$ on it. Is it true that $\mu$ is fully supported if and only if it is ergodic for the action of the shift?

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It seems from this and your previous post that you have some wrong idea in mind of the notion of ergodic. To give a really simple example (the simplest of Vaughn's examples below), imagine a measure supported at the point 0000...... That is $\mu(A)$ is 1 if $A$ contains $000\ldots$ and 0 otherwise. It's not hard to check that $\mu$ is invariant: $A$ contains $0\ldots$, if and only if $S^{-1}A$ contains $0\ldots$. It's also immediate that $\mu$ is ergodic. You have to show that every invariant set has measure 0 or 1. -- But every set has measure 0 or 1. –  Anthony Quas Feb 12 '13 at 22:34