Characteristic Function of Random Variable is Well-Defined

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.

The characteristic function $\phi: \R \to \C$ of $X$ is well-defined.

That is:
 * $\map \phi t = \expect {e^{i t X} }$

exists for all $t \in \R$.

Proof
Let $t \in \R$.

By Modulus of Exponential of Imaginary Number is One:
 * $\cmod {e^{i t X } } = 1$

since $t \map X \omega \in \R$ for all $\omega \in \Omega$.

In particular:
 * $\ds \int \cmod {e^{i t X} } \rd \Pr = \int 1 \rd \Pr = 1$

Thus, by $(3)$ of Characterization of Integrable Functions:
 * $\ds \expect {e^{i t X} } = \int e^{i t X} \rd \Pr$

exists.