Weak containment rigidity for distal actions

We prove that if a measure distal action $α$ of a countable group $Γ$ is weakly contained in a strongly ergodic probability measure preserving action $β$ of $Γ$, then $α$ is a factor of $β$. In particular, this applies when $α$ is a compact action. As a consequence, we show that the weak equivalence class of any strongly ergodic action completely remembers the weak isomorphism class of the maximal distal factor arising in the Furstenberg-Zimmer Structure Theorem.

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