It is a well known fact that if $(\Omega, \mathcal{F}, P)$ is a probability triple and $\{A_i : i < k\}$ is a finite collection subsets of $\Omega$, then there is a $P' \supset P$ and $\mathcal{F'} \supset (\mathcal{F} \cup \{A_i : i < k\})$ such that $(\Omega, \mathcal{F'}, P')$ is a probability triple. In this case, let's say that $P$ can be extended to include $\{A_i : i < k\}$, or that $\{A_i : i < k\}$ can be adjoined to $P$.

I would like to know what is the best, and the worst, we can say when we replace the word "finite" in the statement above with "countable". More precisely, I am asking the following:

- What is the biggest class of probability triples and countable collections (that has been proven in $ZFC$) for which the statement above holds? For example, is it true that if $A \subset \mathscr{P}(\Omega)$ is countable, and the complete subalgebra of $(\mathscr{P}(\Omega), \subset)$ generated by $A$ includes no dense linear order, then any probability measure can be extended to include $A$?
- Modulo large cardinal assumptions, what is the best case scenario consistent with $ZFC$? For example, is it consistent that we can extend any probability measure to include any countable collection of subsets of the sample space?
- Modulo large cardinal assumptions, what is the worst case scenario consistent with $ZFC$? For example, is it consistent that for each uncountable cardinal $\kappa$ there is a probability triple $(\kappa, \mathcal{F}, P)$ such that no countable $S \subset \mathscr{P}(\kappa)$ can be adjoined to $P$?

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