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In this paper we show that the natural action of the symmetric group actingon the product space $\{0, 1 \}^{\mathbb{N}} $ endowed with a symmetric measureis approximately transitive. We also extend the result to a larger class ofprobability measures.

Approximate transitivity of the ergodic action of the group of finite permutations of $\mathbb{N}$ on $\{0, 1 \}^{\mathbb{N}}$