Suppose $\circ:G \times X \to X$ with $X$ countable. There is a measure preserving dynamical system $([0,1]^X, \lambda^X, G)$ where $\lambda$ is the uniform measure on $[0,1]$ and for $(a_x)_{x \in X} \in [0,1]^X$, $g$ takes $(a_x)_{x \in X}$ to $(a_{\circ(g,x)})_{x \in X}$.

If $X$ has a finite orbit $Y$ then there is an invariant measure preserving map from $([0,1]^X, \lambda^X)$ to $([0,1], \lambda)$. For example

$$(a_x)_{x \in X} \mapsto \sum_{y \in Y} a_y \mod 1.$$

If $X$ has no finite orbits will there still always be an invariant measure preserving map from $([0,1]^X, \lambda^X)$ to $([0,1], \lambda)$?

As a concrete example, is there such an invariant measure preserving map when $X = (\mathbb{Z}, S)$ and $G = $Aut$(\mathbb{Z}, S)$, where $S$ is the successor relation (and there is no other structure on $\mathbb{Z}$)?

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