Let $f(\cdot)$ denote a density. Measure theory aside, I know that $A$ and $B$ are conditionally independent given $C$ iff $f(a,b|c)=f(a|c)f(b|c).$ Also, $\int f(a,b|c)f(c) dc = f(a,b)$.
But is it true that $\int f(a,b|c)f(c) dc = \int f(a|c)f(b|c)f(c) dc = f(a)f(b)$? I don't think this is always the case, but I am not sure. If it is not true, are there any conditions under which it would be true?
Generally this is false, but it is true if $A,B$ are conditionally independent given $C$.
Generally this is false, but it is true if $A$ is independent of $C$, and also $B$ is independent of $C$.