Domain vs Co-domain vs Support of a random variable

Solution 1:

A random variable is defined a priori over some probability space, say $(\Omega, \mathscr{F}, \mu).$ So, that is what they mean that "$X$ is defined over some set."

Then, $X:\Omega \to \mathbf{R}$ is a (ordinary) function that is measurable (for every interval $I$, $\{X \in I\} := X^{-1}(I)$ is an event, viz. belongs to $\mathscr{F}$). However, $X$ maps the probability measure $\mu$ from the probabilty space $(\Omega, \mathscr{F}, \mu)$ into $\mathbf{R}$ (and the Borel sets of $\mathbf{R}$, call them $\mathscr{B}_\mathbf{R}$). In other words, $X$ induces the new measure $F_X:\mathscr{B}_\mathbf{R} \to \mathbf{R}$ from the original measure $\mu$ by setting $F_X(I) = \mu(X \in I).$ This $F_X$ is known as "distribution function of $X$" by the mathematicians and as "c.d.f. of X" (for "cumulative distribution function") by first year students and the statistician, although, admittedly, the "c." is redundant and unnecessary. Note that we can, at this point, discard $X$ and define a new random variable, say $X_{\mathrm{new}},$ on the probability space $(\mathbf{R}, \mathscr{B}_\mathbf{R}, F_X)$ by $X_{\mathrm{new}}:t \mapsto t$ (i.e. $X_\mathrm{new}$ is just the identity function). Then $X_\mathrm{new}$ also has distribution function $F_X.$ Thus, we can discard the original space where $X$ was defined and assume it was $\mathbf{R}$ all along; since the new $X$ is the identity function, questions over its domain and range are equivalent (and become ambiguous). Note that probabilistically, nothing is lost since $X$ is really just a device to transport $\mu$ from the "abstract probability space" onto the "Borel sets of $\mathbf{R}$."

The support of $X$ is short-hand for the support of the measure $F_X$ and is then the complement of largest open set $O$ such that $F_X(O) = 0$ (i.e. is the smallest closed set where all the "mass of $X$" is concentrated). (Note that I used "mass of $X$" for this is short-hand for a more correct yet pendantic and cumbersome expression: "the mass assigned by $\mu$ on preimages by $X$ of Borel sets in $\mathbf{R}$.") I am a formally trained probabilist and questions about the "range of a random variable" usually never occur as this concept is admittedly ambiguous for random variables.