Let $X$ be an integrable random variable defined on the space $(\Omega , B, P)$ and suppose $G \subset B$ is a sub-sigma field. Prove that on every non-null atom $\Lambda$ of $G$, the conditional expectation $E(X\mid G)$ is constant and
$$E(X\mid G)(\omega) = \int_\Lambda X \, dP/P(\Lambda), \quad \omega \in \Lambda$$
Recall that $\Lambda$ is a non-null atom of $G$ if $P(\Lambda) > 0$ and $\Lambda$ contains no subsets belonging to $G$ other than $\emptyset$ and $\Omega.$
Suppose $Y$ is a conditional expectation, then it must be $G$ measurable.
Suppose $\Lambda$ is a non null atom and $\omega_0 \in \Lambda$. Let $y_0 = Y(\omega_0)$, then since $Y^{-1}\{y_0\} \cap \Lambda \subset \Lambda$ is measurable and non empty, we must have $\Lambda \subset Y^{-1}\{y_0\}$. Hence if $\omega \in \Lambda$, we must have $Y(\omega) = y_0$, in particular, it is constant.
Then we have $\int_\Lambda X dP = \int_\Lambda Y dP = y_0 P(\Lambda)$ and so $Y(\omega) = y_0 = { 1\over p(\Lambda) } \int_\Lambda X dP $ for $\omega \in \Lambda$.