Weak containment rigidity for distal actions

We prove that if a measure distal action $\alpha$ of a countable group $\mathrm{\Gamma }$ is weakly contained in a strongly ergodic probability measure preserving action $\beta$ of $\mathrm{\Gamma }$, then $\alpha$ is a factor of $\beta$. In particular, this applies when $\alpha$ 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.