Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a
D1159: Probability space such that
| (i) |
$E_0, E_1, E_2, \ldots \in \mathcal{F}$ are each an D1716: Event in $P$
|
| (ii) |
$E_0, E_1, E_2, \ldots$ is a D5143: Set partition] of $\Omega$
|
| (iii) |
$\mathcal{G} = \sigma \langle E_0, E_1, E_2, \ldots \rangle$ is a D318: Generated sigma-algebra on $\Omega$ with generators $E_0, E_1, E_2, \ldots$
|
| (iv) |
$X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $P$
|
| (v) |
\begin{equation}
\mathbb{E} |X| < \infty
\end{equation}
|
Let $n \in \mathbb{N}$ be a
D996: Natural number such that
| (i) |
\begin{equation}
\mathbb{P}(E_n) > 0
\end{equation}
|
Then
\begin{equation}
\forall \, \omega \in \Omega
\left( \omega \in E_n \quad \implies \quad \mathbb{E}(X \mid \mathcal{G})(\omega) = \frac{\mathbb{E}(X I_{E_n})}{\mathbb{P}(E_n)} \right)
\end{equation}
