This is an embarrassingly simple question: If $X$ is a normally distributed random variable, what is the support of $Y=X^2$? It clearly should be the positive real line. However, I cannot find a clear definition in any textbook that clearly settles this question and gives this answer.
A point $x$ belongs to the support of a RV if every open neighbourhood of $x$ has positive measure. Going by this definition, the support of $Y$ is $[0,\infty)$: the non-negative real line.