# 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.

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Thank you. I am almost sure this is the correct definition. But where can I find a reference for that definition that applies directly to a random variable? (I find definitions of the support of a measure but nothing definitive seems to apply straight to random variables.) –  Stephan Feb 9 '13 at 3:27
@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
@guy Yes, that would be another natural (and equivalent) definition of the support of a random variable. But do you know of an explicit definition? Do you have a reference? –  Stephan Feb 9 '13 at 15:19
I do not have a reference. I went from the notion of support for a measure. I saw some definitions online [1] gabormelli.com/RKB/Random_Variable_Support and [2] www.stat.umn.edu/geyer/old/5102/n.pdf. [1] is not quite equivalent to the definition I gave. It depends on the underlying measure as @guy points out. [2] is more or less the same. It assumes the density function exists, which may not be general enough. –  Sriram S Feb 13 '13 at 16:39