# Support vs range of a random variable

Is there any difference between the two? I have not met any formal definition of the support of a random variable. I know that for the function $f$ the support is a closure of the set $\{y:\;y=f(x)\ne0\}$.

The support of the probability distribution of a random variable $X$ is the set of all points whose every open neighborhood $N$ has the property that $\Pr(X\in N)>0$.
The complement of the support is the union of all open sets $G$ such that $\Pr(X\in G)=0$. Since the complement is a union of open sets, the complement is open. Therefore the support is closed.
• I see. It is often written "let $X$ be a random variable with support S", which confused me, since the random variable is also a function. Thank you for clarification! – Bayes Jun 10 '13 at 2:33
• Note that the support of the distribution of $X$ is the same as the essential range of the measurable function $X$ itself. – Nate Eldredge Jun 10 '13 at 2:54
• What about measure theory? I believe the definition of the support of a function $f : A \rightarrow B$ is : {$x \in A : f(x) \neq 0$}, that is the same as the range of the function, right? – Frank Aug 13 at 13:15