Let $Z \in \text{Random}(\mathbb{C})$ be a D4877: Random complex number such that

(i)	\begin{equation} \exists \, C \in (0, \infty) : |Z| \overset{a.s.}{\leq} C \end{equation}

Let $p \in [0, \infty)$ be an D4767: Unsigned real number.

Since $|Z| \leq C$ almost surely, result R4127: Real exponentiation function with unsigned exponent is isotone on unsigned reals implies that $|Z|^p \leq C^p$ almost surely. Applying result R1818: Isotonicity of real expectation, we thus get \begin{equation} \mathbb{E} |Z|^p \leq \mathbb{E} C^p = C^p < \infty \end{equation}