I have a basic question regarding the definition of a random variable. Probability and Random Processes (Grimmett and Stirzaker) have the following:
A random variable is a function $X:\Omega\rightarrow \mathbb{R}$ with the property that
$
\{
\omega\in \Omega: X(\omega)\leq x
\}
\in \mathcal{F}
$
for each $x\in \mathbb{R}$. Such a function is said to be $\mathcal{F}$-measurable.
Q1: Because of the curly brackets I guess $\{\omega\in \Omega: X(\omega)\leq x\}$ is a set, right?
Q2: I know if $a$ is an element of the set $A$ we write $a\in A$. But if a set $B$ is a subset of a set $C$, we write $B\subset C$ and not $B\in C$.
So if $\{\omega\in \Omega: X(\omega)\leq x\}$ is a set shouldn't we use "$\subset$" instead of "$\in $", i.e.
$$
\{
\omega\in \Omega: X(\omega)\leq x
\}
\subset \mathcal{F} \qquad ? \tag 1
$$
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