Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) | $E \in \mathcal{F}$ is an D1716: Event in $P$ |
(ii) | \begin{equation} \mathbb{P}(E) > 0 \end{equation} |
(iii) | $\mathcal{G} = \sigma \langle E \rangle$ is a D318: Generated sigma-algebra on $\Omega$ with generator $E$ |
(iv) | $X : \Omega \to \mathbb{R}$ is an D3161: Random real number on $P$ |
(v) | \begin{equation} \mathbb{E} |X| < \infty \end{equation} |
Then
\begin{equation}
\forall \, \omega \in \Omega
\left( \omega \in E \quad \implies \quad \mathbb{E}(X \mid \mathcal{G}) = \frac{\mathbb{E}(X I_E)}{\mathbb{P}(E)} \right)
\end{equation}