The non-ergodic Host-Kra-Ziegler structure theorem for $\mathbb{Z}^d$-actions via measurable selections

We establish a non-ergodic version of the Host-Kra-Ziegler structure theorem for measure-preserving $\mathbb{Z}^d$-actions. Our argument reduces the non-ergodic case to the ergodic theorem (for $d\ge 2$ due to Candela and Szegedy) via a measurable selection procedure.

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