Suppose that we have a sample space $\Omega$ and an event, $A \subseteq \Omega$. The indicator function, which is a random variable, is defined as follows:
$\mathbb{1}_A(\omega) = \begin{cases}1 & \text{if } \omega \in A \\
0 & \text{if } \omega \notin A
\end{cases}$
where $\omega$ is a possible outcome in $\Omega$. The support (or the possible values that a random variable can take) of this indicator random variable is $\{0, 1\}$.
The probability mass function is as follows:
$p_{\mathbb{1}_A}(x) = \begin{cases} P(A) & \text{if } x = 1 \\
1-P(A) & \text{if } x = 0 \\
0 & \text{otherwise}
\end{cases}$
Then, the expected value of the indicator random variable is:
$\mathbb{E}[\mathbb{1}_A] = \displaystyle\sum_{x \in \{0,1\}}x \cdot p_{\mathbb{1}_A}(x) = P(A)$
As a side, remember that a real valued random variable is a function. It maps the sample space, $\Omega$, into the real numbers, $\mathbb{R}$. Mathematically, it is $X:\Omega \rightarrow \mathbb{R}$, where $X$ is a real valued random variable.
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