Question: 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.
$$
\{
\omega\in \Omega: X(\omega)\leq x
\}
\subset \mathcal{F} \qquad ? \tag 1
$$

Answer: 
Q1: Yes, that is a set.

Q2: The thing you need to understand here is that $\mathcal{F}$ is a collecion of sets. In other words, it is a set of sets - its elements are sets. Hence we use $\in$ instead of $\subset$.

For example,

$$1 \in \{1, 2, 3, 4\}$$
$$\{1\} \subset \{1, 2, 3, 4\}$$
but
$$\{1\} \in \{\{1\}, \{2\}, \{3\}, \{4\}\}\text{.}$$
Q1: Yes, this is a set of points in your probability space.
Q2: $\mathcal{F}$ is actually a set of subsets of your probability space (called a sigma algebra) This set defines events that you are allowed to work with. The reason is that for measure theoretic reasons we generally can't define a good probability measure on all subsets. Therefore your last statement is simply that the event $X(\omega) \leq x$ is in your sigma algebra.