My book has the following definition for the range of a random variable.
Since a random variable is defined on a probability space, we can calculate these probabilities given the probabilities of the sample points. Let $a$ be any number in the range of a random variable $X$. Then the set $\{\omega \in \Omega :X(\omega)=a\}$ is an event in the sample space (simply because it is a subset of $\Omega$).
I think I'm misunderstanding the provided notation but to me this expression:
$$\{\omega \in \Omega :X(\omega)=a\}$$
seems to be saying "For each sample point $\omega$ in the sample space $\Omega$, compute $X(\omega)$ and store the result $a$ in the set". If the above expression is saying what I think it says how can we say that a set of $\{X(\omega_1), X(\omega_2),\ldots,X(\omega_{|\Omega|})\}$ is an event in the sample space? Aren't events a subset $\{ \omega_1, \omega_2, \ldots\}$ of $\Omega$ ?
Best Answer
$\{\omega \in \Omega :X(\omega)=a\}$ means the sets of points of $\Omega$ such that every point is mapped to the value of $a$ by the random variable $X: \Omega \mapsto \mathbb R$.
$\{X(\omega_1), X(\omega_2),\ldots,X(\omega_{|\Omega|})\}$ is better to be noted as $X(\Omega)$, as $\omega_{|\Omega|}$ sometimes does not make sense if $\Omega$ is not finite. But you are right that it is not an event, because it is a subset of $\mathbb R$.
But some times we write $X$ directly as a presentation/notation. For example, we write $u<X\leq v$ as a short form to represent the event of $\{\omega \in \Omega: u < X(\omega) \le v\}$. We always focus on subsets of $\Omega$ as events, because the probability is defined over the measurable subsets of $\Omega$, which is in a $\sigma$-algebra.