Although I have already discussed this point repeatedly on this ‘Og, I found myself replying to [yet] another question on X validated about the apparent paradox of conditioning on a set of measure zero, as for instance when computing
P(X=.5 | |X|=.5)
which actually has nothing to do with Bayesian inference or Bayes’ Theorem, but is simply wondering about the definition of conditional probability distributions. The OP was correct in stating that
P(X=x | |X|=x)
was defined up to a set of measure zero. And even that
P(X=.5 | |X|=.5)
could be defined arbitrarily, prior to the observation of |X|. But once |X| is observed, say to take the value 0.5, there is a zero probability that this value belongs to the set of measure zero where one defined
P(X=x | |X|=x)
arbitrarily. A point that always proves delicate to explain in class…!
