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}

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

(i)	$G \in \mathcal{F}$ is an D1716: Event in $P$
(ii)	\begin{equation} \mathbb{P}(G) > 0 \end{equation}
(iii)	$X : \Omega \to \mathbb{R}$ is an D3161: Random real number on $P$
(iv)	\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 G) = \frac{\mathbb{E}(X I_G)}{\mathbb{P}(G)} \right) \end{equation}