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Let $X,Y,Z$ be random variables jointly distributed on the same probability space then we know the conditional probability $P(X=a,Y=b\mid Z)$ is a random variable which is $Z$ measurable. Can we say
$\begingroup$That last is one of the related things that are sometimes called the law of total probability. It can be stated by saying that the prior expected value of the posterior probability is equal to the prior probability. $$\operatorname E( \Pr(A\mid Z)) = \Pr(A). $$$\endgroup$
$\begingroup$The problem with your first question is: How do you define $P(X = a ~|~ Y = b,Z)$? If you define it as $P(X = a, Y = b|Z) / P(Y = b|Z)$, then the answer is of course yes. I don't think this is what you want. Otherwise it seems more simple and clear to work with conditional expectations with respect to some sub-$\sigma$-algebras $\cal{G} \subset \cal{H}$.$\endgroup$