We assume in this question that $X ( \Omega )$ is a countable set. We denote $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ where the $x _ { n }$ are pairwise distinct. For all $n \in \mathbb { N }$, we set $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$.\\
Show that $\phi _ { X }$ is defined on $\mathbb { R }$ and that, for all real $t , \phi _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \mathrm { e } ^ { \mathrm { i } t x _ { n } }$.