Conditional Expectation Exists and is Almost Surely Unique
# Statement
Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space and let $X$ be a Random Variable on $\Omega$. Let $\mathcal{G} \subset \mathcal{B}$ be a sub-Sigma Algebra. Then there exists a Conditional Expectation $Y$ for $X$ with respect to $\mathcal{G}$. If $Y’$ is another such Conditional Expectation then $$Y = Y’ \text{ almost surely}$$
Therefore, it makes sense to refer to $\mathbb{E}[X|\mathcal{G}]$ as the Conditional Expectation of $X$ with respect to $\mathcal{G}$.
# Proof
Write for $A \in \mathcal{G}$
$$\nu(A) = \int\limits_{A} X d \mathbb{P}(\omega)$$
Then, because Integration defines an Absolutely Continuous Measure, we know that $\nu$ is a Signed Measure on $\mathcal{G}$ and $\nu « \mathbb{P} {\big|}{\mathcal{G}}$. Furthermore, $\mathbb{P}$ is a Finite Measure and thus a Sigma-Finite MeasureThus, there exists a $\mathbb{P}$-Almost Surely $\mathcal{G}$-measureable unique Radon-Nikodym Derivative $Z = \frac{d\nu}{d \mathbb{P} {\big|}{\mathcal{G}}}$ so that for $A \in \mathcal{G}$
$$\nu(A) = \int\limits_{A} Z d \mathbb{P}(\omega) = \int\limits_{A} X d \mathbb{P}(\omega)$$
Thus $Z$ satisfies the definition of Conditional Expectation. Since $Z$ is $\mathbb{P}$-Almost Surely unique, any other $\mathcal{G}$-measureable Random Variable satisfying the definition of Conditional Expectation is also a Radon-Nikodym Derivative and is thus Almost Surely $Z$. $\blacksquare$