Since $X$ takes value $a$ with probability one, then the random variable $e^{i t \cdot X}$ takes value $e^{i t \cdot a}$ with probability one. Thus, applying
R1814: Expectation of discrete random euclidean real number, we have
\begin{equation}
\mathbb{E}(e^{i t \cdot X}) = e^{i t \cdot a} \mathbb{P}(X = a) = e^{i t \cdot a}
\end{equation}
$\square$
