I'm a number theorist finding myself needing to use some concepts from probability that are probably (no pun intended) quite basic to experts; I would rather cite a readily available source than reinvent the wheel myself.
Specifically, I would like to find a source (monograph/graduate textbook, perhaps) that gives a definition of a "strictly positive" random variable (one that assigns positive probability to every nonempty open set - not one that takes values in \mathbb R_{>0}). Ideally, I would like to find a source that already contains a proof of the following lemma: if $X$ and $Y$ are independent random variables taking values in the same space ($\mathbb R^n$, say), and if $X$ is strictly positive, then $X+Y$ is also strictly positive.