Let $(\mathrm{\Sigma},F,P)$ be a probability space, let $X$ be an $F$-random variable, let $G$ be a sub-sigma algebra of $F$, let $\omega \in \mathrm{\Omega}$, let $A$ be the intersection of all the sets in $G$ that contain $\omega $, and suppose that $A\in F$. Then is it true that $E(X|G)(\omega )=E(X|A)$?

I just wrote this statement down based on my intuition of what conditional expectation means, and I want to verify if it’s actually true. Note that sigma algebras need not be closed under uncountable intersections, so $A$ need not be an element of $G$. But I assumed $A$ is at least an element of $F$ so that $E(X|A)$ is meaningful.

I just wrote this statement down based on my intuition of what conditional expectation means, and I want to verify if it’s actually true. Note that sigma algebras need not be closed under uncountable intersections, so $A$ need not be an element of $G$. But I assumed $A$ is at least an element of $F$ so that $E(X|A)$ is meaningful.