Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) | $\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$ |
(ii) | $X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $P$ |
(iii) | \begin{equation} \mathbb{E} |X| < \infty \end{equation} |
(iv) | \begin{equation} \sigma_{\text{pullback}} \langle X \rangle \subseteq \mathcal{G} \end{equation} |
Then
\begin{equation}
\mathbb{P} \left( \mathbb{E}(X \mid \mathcal{G}) = X \right)
= 1
\end{equation}