# Questions about regular conditional probabilities

In Section 9.9. Regular conditional probabilities and pdfs of David Williams' Probability and Martingales:

By linearity and (cMON), we can show that for a fixed sequence $(F_n)$ of disjoint elements of $\mathcal{F}$, we have $$(a) P(\cup F_n | \mathcal{G}) = \sum P(F_n | \mathcal{G})$$ a.s.. Except in trivial cases, there are uncountably many sequences of disjoint sets, so we cannot conclude from (a) that there exists a map $$P(.,.) : \Omega \times \mathcal{F} \to [0,1)$$ such that

(b1) for $F \in \mathcal{F}$, the function $P(.,F)$ is a version of $P(F| \mathcal{G})$;

(b2) for almost every $\omega$, the map $P(\omega, .)$ is a probability measure on $\mathcal{F}$.

I wonder why "there are uncountably many sequences of disjoint sets, so we cannot conclude from (a) that ..."?

How do "a fixed sequence" or "uncountably many sequences" of disjoint sets have different effect on whether "$P(\omega, .)$ is a probability measure on $\mathcal{F}\,$", given that a measure is defined to be additive for countably many disjoint sets?

Thanks and regards!

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Because each (a) holds for every $\omega$ not in a negligible set but, for a given $\omega$, (b2) may fail as soon as (a) fails for at least one sequence $(F_n)$, that is, as soon as $\omega$ is in any one of these negligible sets. And the union of these negligible sets may be non negligible since there are uncountably many sequences $(F_n)$. – Did Nov 8 '11 at 8:20