conditioning on continuous value

Question

Is there a formula like this $$P(A)=\int P(A|x)p(x)dx $$ where $x$ is a continuous random variable? If yes, what is a good book for learning about this? Thanks!

Answer 1

Yes, there is such a result (or, rather, this is the defining equation of the conditional probability $\mathbb{P}(A|X))$. However, this really naturally belongs to the theory of general conditional expectations, which is firmly measure theoretic, so that is probably a bit far away if you aren't familiar with any measure theory.

René Schilling's Measures, Integrals & Martingales contains both an introduction to measure theory (which is very good) and an introduction to conditional expectations and (as the title hints) some martingale theory (which is one of the best starting points to get acquainted with how conditional expectations work). He doesn't devote a lot of time to regular probability distributions, though.