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Let X $X$ be a separable complete metric space and Z $Z$ be the set of all integers. Let nu $\nu$ be a Borel probability measure on X^Z $X^Z$ invariant under the shift function $S:X^Z --> X^Z\to X^Z$. Is it necessarily the case that nu $\nu = mu^Z \mu^Z$ for some Borel probability measure mu $\mu$ on X?

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Product Measure Only Possible Measure?

Let X be a separable complete metric space and Z be the set of all integers. Let nu be a Borel probability measure on X^Z invariant under the shift function S:X^Z --> X^Z. Is it necessarily the case that nu = mu^Z for some Borel probability measure mu on X?

Thanks