Good reference for strictly positive random variables 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.
 A: The following is Proposition 2.1.3 (page 23) in Werner Linde's book 
Probability in Banach Spaces - Stable and Infinitely Divisible Distributions.
It should give you what you need when $X$ and $Y$ are independent. Let $\mu$ be the distribution of $X$ and $\nu$ the distribution of $Y$, so that $\mu*\nu$ is the distribution of $X+Y$. 

If $\mu,\nu$ are Radon measures on the Borel sets ${\cal B}(E)$ of a Banach space $E$,
  then $$\mbox{supp}(\mu*\nu)=\overline{\mbox{supp}(\mu)+\mbox{supp}(\nu)}.$$ 

For a proof, the reader is referred to Theorem 1.2.1 of Probability Measures on Locally Compact Groups by H. Heyer. I think that this would be a good, standard reference, though I don't own Heyer's book so I can't check it. 
A: No reference, but here goes, assuming that $X$ and $Y$ are independent and take values in a finite-dimensional real vector space:
It is enough to prove the property for an arbitrary open ball $B_r(x)$. The entire value space is covered by countably many open balls of radius $r/2$; by countable additivity at least one of these balls, call it $B_{r/2}(y)$, must contain a positive probability mass for $Y$.
Now, by assumption $P(X\in B_{r/2}(x-y))$ is also positive, and $P(X+Y\in B_r(x))$ must be at least the product of these probabilities.
