Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $E$. We call the distribution of the variable $X$ and denote by $\mu_X$ the map $$\begin{array}{rcl} \mu_X : & \mathscr{P}(E) & \rightarrow [0;1] \\ & A & \mapsto P(\{X \in A\}) \end{array}$$ where $\{X \in A\} = \{\omega \in \Omega \text{ such that } X(\omega) \in A\}$. Verify that $\mu_X$ is a probability on $E$.
Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $E$. We call the distribution of the variable $X$ and denote by $\mu_X$ the map
$$\begin{array}{rcl} \mu_X : & \mathscr{P}(E) & \rightarrow [0;1] \\ & A & \mapsto P(\{X \in A\}) \end{array}$$
where $\{X \in A\} = \{\omega \in \Omega \text{ such that } X(\omega) \in A\}$.
Verify that $\mu_X$ is a probability on $E$.