Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
| (i) | $Z : \Omega \to \mathbb{C}$ is a D4877: Random complex number on $P$ |
| (ii) | \begin{equation} \mathbb{E} |Z| < \infty \end{equation} |
| (iii) | $\xi : \Omega \to \Xi$ is a D202: Random variable on $P$ |
| (iv) | $\{ \xi = s_0 \}, \{ \xi = s_1 \}, \{ \xi = s_2 \}, \ldots \in \mathcal{F}$ are each an D1716: Event in $P$ |
| (v) | $\{ \xi = s_0 \}, \{ \xi = s_1 \}, \{ \xi = s_2 \}, \ldots$ is a D83: Proper set partition of $\Omega$ |
| (vi) | \begin{equation} \forall \, n \in \mathbb{N} : \mathbb{P}(\xi = s_n) > 0 \end{equation} |
Then
\begin{equation}
\mathbb{E}(Z)
= \sum_{n = 0}^{\infty} \mathbb{E}(Z \mid \xi = s_n) \mathbb{P}(\xi = s_n)
\end{equation}