# Definition of the Support of a Real Valued Random Variable

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.

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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.
@Stephan The support of an rv should be defined to be the support of the induced measure on $\mathbb R$. –  guy Feb 9 '13 at 4:51