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  • $\begingroup$ The final $\sigma$-algebra $\{M\subseteq \frac{K_X}{\sim}: \pi^{-1}(M)\in \mathcal B(K_X)\}$? $\endgroup$
    – Jochen Wengenroth
    Commented Mar 1, 2023 at 14:43
  • $\begingroup$ @JochenWengenroth Is that standard? (Indeed, another equivalent way of formulating my question is whether the final $\sigma$-algebra admits a sub-$\sigma$-algebra that is standard.) $\endgroup$
    – Julian Newman
    Commented Mar 1, 2023 at 14:48