Conditional expectation is a class not a canonical pointwise function

Under the AC existence convention, identities between conditional expectations concern almost-sure classes. Once versions for finitely or countably many such identities are selected, the identities hold simultaneously outside the union of their measurable null exceptional sets; that union is still null. This countable-union argument gives no general guarantee that one chosen representative satisfies an uncountable family simultaneously, although particular uncountable families may admit such a representative. Subsequent conditional-law constructions require their own hypotheses.

The class and version terminology is Conditional expectation as an ae class, whose existence uses The Axiom of Choice. A modified version must remain measurable for the conditioning sigma-algebra; an arbitrary subset of an ambient null set need not be measurable for that sigma-algebra.

Durrett §4.1 uniqueness discussion, printed p.206, and countable-exception remark after Theorem 4.1.10, p.211; van der Vaart warning after Lemma 1.10, printed p.4.