Conditional events that are not in the event algebra?

Question

The Wikip. page on conditional event algebra states that:

David Lewis showed that in orthodox probability theory, only certain trivial Boolean algebras with very few elements contain, for any given A and B, an event X which satisfies P(X) = P(B|A). Later extended by others, this result stands as a major obstacle to any talk about logical objects that can be the bearers of conditional probabilities.

I haven't read the paper by Lewis. But I'm aware that in Bayesian analysis, the conditional likelihood P(B|A) is not a probability distribution (B here represents data or evidence, while A represents model parameters, see, eg Guyonnet & Ferson "Bayesian methods in risk assessment" p.11) There must be a connection here right?

So I'm primarily interested in Bayesian data analysis but would appreciate an explanation in terms of basic probability theory to relate events and probability measures: What are some simple examples of conditional events that are not in the event algebra?

Do such example involve measure-zero events (which are always problematic)?

Answer 1

in orthodox probability theory, only certain trivial Boolean algebras with very few elements contain, for any given A and B, an event X which satisfies P(X) = P(B|A).

Really? On $(\Omega,\mathcal F,\mathbb P)=([0,1],\mathcal B([0,1]),\mathrm{Leb})$, for every $A$ and $B$, there exists an event $X$ such that $\mathbb P(X)=\mathbb P(B\mid A)$ (as soon as the RHS exists): pick $X=[0,x]$ where $x=\mathbb P(B\mid A)$.

...in Bayesian analysis, the conditional likelihood P(B|A) is not a probability distribution...

Is that so? When it exists, the mapping $\mathbb P(\ \mid A)$ is very much a probability measure.

What are some simple examples of conditional events that are not in the event algebra?

Please define "conditional event".