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

(i)	$X : \Omega \to \mathbb{R}^D$ is a D4383: Random Euclidean real number on $P$
(ii)	$E_0, E_1, E_2, \ldots \in \mathcal{F}$ are each an D1716: Event in $P$
(iii)	$E_0, E_1, E_2, \ldots$ is a D5143: Set partition of $\Omega$

This result is a particular case of R4565: Countable indicator partition of a euclidean real function. $\square$