Suppose we have a discrete probability space $(\Omega,\Sigma,\mathbb{R})$ and a discrete random variable $X:\Omega \to \mathbb{R}$.
A usual way to denote the set of values that $X$ takes is simply using the image notation, $\operatorname{Im}(X)$. For instance, this is useful in defining the iteration variable $x$ when calculating the expectation: $\sum\limits_{x \in \operatorname{Im}(X)}x\cdot\mathbb{P}\{X=x\}$.
I have seen a strange notation in some literature (as well as Proof Wiki) which denote the image of $X$ as $\Omega_X$, and would see this kind of summation for the expectation: $\sum\limits_{x \in \Omega_X}x\cdot\mathbb{P}\{X=x\}$
I cannot quite understand the rationale behind $\Omega_X$ notation to describe the image; it is definitely not a subset of $\Omega$. I have never seen the image of some function $f:A\to B$ denoted as $A_f$, but authors of probability texts do strange things.
My questions are two-fold
- What is a meaningful rationale for this notation?
- Is this standard notation, and is its use encouraged in mathematics literature?
Sometimes one uses the notation $\operatorname{supp}(X)$, the support of $X$, or the support of the probability distribution of $X$, to mean the closure of the set of all points in $\mathbb R$ for whose every open neighborhood the probability that $X$ is in that neighborhood is positive. For a discrete distribution, that is simply the set of all values that the random variable can attain.
I agree that the notation $\Omega_X$ is infelicitous.