Conditional Expectation of a Square-Integrable Random Variable

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$, and let $X$ be a square-integrable random variable on it. Write $\mathbf{1}_{A}$ for the function equal to $1$ on $A$ and $0$ off $A$.

Definition. A random variable $Y$ on $(\Omega,\mathcal{F},P)$ is a conditional expectation of $X$ given $\mathcal{G}$ if:

(i) $Y$ is $\mathcal{G}$-measurable, that is, $Y^{-1}(B)\in\mathcal{G}$ for every Borel set $B$;

(ii) $Y$ is square-integrable;

(iii) $\mathbb{E}[X\mathbf{1}_{A}]=\mathbb{E}[Y\mathbf{1}_{A}]$ for every $A\in\mathcal{G}$, the expectations being defined because the products are integrable by Square-Integrable Random Variables and the Mean-Square Inner Product.

By Existence and Uniqueness of Conditional Expectation for Square-Integrable Random Variables, a conditional expectation of $X$ given $\mathcal{G}$ exists; it is equivalently characterized among $\mathcal{G}$-measurable square-integrable random variables as the best mean-square approximation of $X$, or by the orthogonality relation of that theorem; and any two conditional expectations $Y,Y'$ of $X$ given $\mathcal{G}$ satisfy $P(Y=Y')=1$.

Notation. The symbol $\mathbb{E}[X\mid\mathcal{G}]$ denotes a conditional expectation of $X$ given $\mathcal{G}$. Formally, every statement containing the symbol $\mathbb{E}[X\mid\mathcal{G}]$ is an abbreviation for the assertion that the statement holds for every random variable $Y$ that is a conditional expectation of $X$ given $\mathcal{G}$ in the sense above. (No canonical choice is made, no quotient space is formed, and $\sigma$-algebras are not completed; whenever a specific representative matters, it is named explicitly.)

Two immediate consequences. Taking $A=\Omega$ in (iii): every conditional expectation satisfies $\mathbb{E}\bigl[\mathbb{E}[X\mid\mathcal{G}]\bigr]=\mathbb{E}[X]$. If $X$ is itself $\mathcal{G}$-measurable, then $Y=X$ satisfies (i)-(iii), so $P\bigl(\mathbb{E}[X\mid\mathcal{G}]=X\bigr)=1$, meaning that every conditional expectation of $X$ given $\mathcal{G}$ equals $X$ with probability $1$, by the uniqueness assertion of Existence and Uniqueness of Conditional Expectation for Square-Integrable Random Variables.