Definition 5.3.1. Probability "Mass" Function.
Given a discrete random variable 5.2.1 \(X\) on a space \(R\text{,}\) a probability mass function on \(X\) is given by a function \(f:R \rightarrow \mathbb{R}\) such that:
\begin{align*}
& \forall x \in R , f(x) \gt 0\\
& \sum_{x \in R} f(x) = 1\\
& A \subset R \Rightarrow P(X \in A) = \sum_{x \in A}f(x)
\end{align*}
For \(x \not\in R\text{,}\) you can use the convention \(f(x)=0\text{.}\)