In David Williams's Probability with Martingales, there is a remark regarding conditional expectation of a random variable conditional on a $\sigma$-algebra:
The 'a.s.' ambiguity in the definition of conditional expectation is something one has to live with in general, but it is sometimes possible to choose a canonical version of $E(X| \mathcal{Q})$.
What is "canonical version of $E(X| \mathcal{Q})$", and what are some cases when it is possible to choose it?
I don't want to be misleading, but is it referring to elementary definitions of conditional distribution and conditional expectation when they exist i.e. when the denominators are not zero?
Thanks and regards!
If $E(X|\mathcal{Q})$ is equal (a.s.) to a continuous function, then the continuous function would be a canonical version.
Assume that $\mathcal Q=\sigma(Z)$ for some real valued random variable $Z$, then $E(X\mid\mathcal Q)=u(Z)$ almost surely, for a given measurable function $u:\mathbb R\to\mathbb R$, as well as for every other measurable function $v$ such that $u=v$ $P_Z$-almost everywhere. If one of these functions $v$ is, say, continuous, then $v(Z)$ might be called a canonical version of $E(X\mid\mathcal Q)$.
Unfortunately, this is a dubious denomination since it may well happen that $\mathcal Q=\sigma(Z')$ for a quite different real valued random variable $Z'$. Even if $E(X\mid\mathcal Q)=v'(Z')$ almost surely, for a given continuous function $v'$, nothing ensures that $v(Z)=v'(Z')$ everywhere. One only knows that $v(Z)=v'(Z')$ almost surely and one is back at square one, which is that there is no way to decide which random variable $v(Z)$ or $v'(Z')$ is more canonical than the other...