Probability Notation: What does $\{\omega\in \Omega : X(\omega) \in A\}$ mean? In the book Stats with Julia on p. 79 is reads ...

"The probability distribution of a random variable fully describes the
  probabilities of the events such as $\{\omega\in \Omega : X(\omega) \in A\}$ for all sensible $A \subset R$"

How would you say "$\{\omega\in \Omega : X(\omega) \in A\}$" in plain English?
Is it ....
for every possible outcome $(\omega)$ in the $(\in)$ event space $(\Omega)$ such that $(:)$ there is some specific outcome $(X(\omega))$ in the set $A$ where set $A$ contains real numbers
.. is that close??
 A: We have 


*

*$\Omega$ = outcome space.

*$\omega$ = a particular outcome (that is, $\omega \in \Omega$).

*If $Z$ is an event then it is a subset of $\Omega$ (that is, $Z \subseteq \Omega)$. (*See footnote for an additional detail.)
Indeed the random variable $X$ is a function $X:\Omega \rightarrow \mathbb{R}$. Suppose $A$ is some given subset of real numbers.  Then the following is a subset of $\Omega$: 
$$  \{\omega \in \Omega : X(\omega) \in A\} $$
We interpret this as:  
\begin{align} 
\{\cdot\} \quad  &= \quad \mbox{"The set of ..."}\\
\omega \in \Omega \quad &= \quad \mbox{"outcomes $\omega$ in the outcome space $\Omega$...}" \\
: \quad &= \quad \mbox{"such that..."}\\
X(\omega) \in A \quad  &= \quad \mbox{"$X(\omega)$ is in the set $A$"}
\end{align}
Put all together  it reads:  
The set of outcomes $\omega$ in the outcome space $\Omega$ such that $X(\omega)$ is in the set $A$.
Notice that 
$$ \{\omega \in \Omega : X(\omega) \in A\} \subseteq \Omega$$

Example: 
\begin{align}
\Omega &= \{blue, red, green, pink\}\\
X(blue) &= 2\\
X(red) &= 2.5\\
X(green) &=0\\
X(pink) &=-3\\
A &= \{2, -3, 8\}\\
B &= \{2.5, 0, -3\}\\
C &= \{x \in \mathbb{R} : x\leq 1\} = (-\infty, 1]
\end{align}
Then 
\begin{align}
\{\omega \in \Omega : X(\omega) \in A\} &= \{blue, pink\}\\
\{\omega \in \Omega : X(\omega) \in B\} &= \{red, green, pink\}\\
\{\omega \in \Omega : X(\omega) \in C\} &= \: ??? \quad \quad [\mbox{Exercise}]\\
\{\omega \in \Omega : X(\omega) \in A \cap B\} &= \: ???\quad \quad [\mbox{Exercise}]\\
\{\omega \in \Omega : X(\omega) \notin A\} &= \: ???\quad \quad [\mbox{Exercise}]\\
\{\omega \in \Omega : X(\omega) > 0\} &= \: ??? \quad \quad[\mbox{Exercise}]\\
\{\omega \in \Omega : X(\omega) \leq 0\} &= \: ??? \quad \quad[\mbox{Exercise}]\\
\{\omega \in \Omega : X(\omega) \leq 100\} &= \: ??? \quad \quad[\mbox{Exercise}]\\
\{\omega \in \Omega : X(\omega) \leq -78\} &= \: ??? \quad \quad[\mbox{Exercise}]
\end{align}
How many possible events are there (for this example)? 
A: [Edit] Yeah. The way to see, is to look to $X$ being a function (random variable).  $$X: \Omega \longrightarrow \mathbb{R}.$$
The subset $A\subset \Bbb R$ is just to know what values can be and restrict the events $\omega\in \Omega$.
