Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X_0, X_1, X_2, \ldots : \Omega \to [0, \infty]$ are each a D5101: Random unsigned basic number on $P$

Then \begin{equation} \mathbb{E} \left( \sum_{n \in \mathbb{N}} X_n \right) = \sum_{n \in \mathbb{N}} \mathbb{E}(X_n) \end{equation}

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X_0, X_1, X_2, \ldots : \Omega \to [0, \infty]$ are each a D5101: Random unsigned basic number on $P$

Then \begin{equation} \mathbb{E} \left( \lim_{N \to \infty} \sum_{n = 0}^N X_n \right) = \lim_{N \to \infty} \sum_{n = 0}^N \mathbb{E}(X_n) \end{equation}

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X_0, X_1, X_2, \ldots : \Omega \to [0, \infty]$ are each a D5101: Random unsigned basic number on $P$

Then \begin{equation} \mathbb{E} \left( \sum_{n = 0}^{\infty} X_n \right) = \sum_{n = 0}^{\infty} \mathbb{E}(X_n) \end{equation}