I just realised I've been thinking about conditional expectation in a fallacious way: I've been assuming that the continuous case follows the discrete case in the sense that for probability space $(X,\mu)$, and measurable $T: X\rightarrow \mathbb R$, the function $$L^1(X,\mu) \rightarrow \mathbb R,~~~~ f\mapsto \mathbb E_\mu[~f~| ~t ~] $$ is a measure for $T_\ast \mu$-almost every $t\in \mathbb R$. But of course this doesn't really make sense since in order to define $\mathbb E_\mu[~f~| ~t ~]$ you have to take a radon Nikodym derivative that depends on $f$, and that derivative is only defined up to a set of measure zero. So we have a possibly different null set for every $f\in L^1$.
Presumably this factorisation can be rescued in some way with the right assumptions, though. What's the best way to do that? I'm particularly interested in the case where $T$ is a sufficient statistic for a family of $\mu_\theta : \theta \in \Theta$, so then there are even more annoying null sets if one wants to make a family of measures $$\{f\mapsto \mathbb{E}_{\Theta}[f,t]: t\in \mathbb{R}\}.$$