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Result R5305 on D2280: Conditional expectation

Conditional expectation of a random complex matrix given maximal information

Formulation 0

Let

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

(i)	$Z : \Omega \to \mathbb{C}^{N \times M}$ is a D5884: Random complex matrix on $P$
(ii)	$\begin{equation} \mathbb{E} \|Z\| < \infty \end{equation}$

Then

$\begin{equation} \mathbb{E}(Z \mid \mathcal{F}) \overset{a.s.}{=} Z \end{equation}$