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 an D3161: Random real number on $P$
(iii)	\begin{equation} \mathbb{E} |X| < \infty \end{equation}
(iv)	$\sigma_{\text{pullback}} \langle X \rangle, \mathcal{G}$ is an D471: Independent collection of sigma-algebras in $P$

We show that $\mathbb{E} X$ is the conditional expectation of $X$ given $\mathcal{G}$ by confirming that it satisfies the required properties. To start off, result R1177: Constant map is always measurable guarantees that $\mathbb{E} X \in \mathcal{G}$.

Next, fix $G \in \mathcal{G}$ and let $I_G$ denote the indicator function for $G$ within $\Omega$. Since $X$ and $\mathcal{G}$ are independent, result R2524: Independent sigma-algebra produces independent indicator functions shows that so are $X$ and $I_G$. Thus, we may apply result results

(i)	R2258: Expectation of product is multiplicative for independent random numbers
(ii)	R4652: Real-linearity of real expectation

to conclude \begin{equation} \mathbb{E}(X I_G) = \mathbb{E} X \mathbb{E} I_G = \mathbb{E}(\mathbb{E}(X) I_G) \end{equation} $\square$