My instructor defined $\mathbb E(X|\mathcal G)$ in the usual way and mentioned that it can also be characterized as an integral of $X$ with respect to some measure. Similarly to $\mathbb E(X|A)=\int X \ d\mathbb P_A$ where $\mathbb P_A:=\mathbb P(\cdot|A)$.
I couldn't find anything of this sort in my textbooks.
What could it possibly mean?
An added clarification of the notation: We are in $(\Omega, \mathcal F, \mathbb P)$, $\mathcal G$ is a sub-$\sigma$-algebra of $\mathcal F$ and $A\in \mathcal F$ is an event with $\mathbb P(A)>0$.
When the sigma-algebra $G$ is discrete and generated by a partition $(A_n)$ of $\Omega$ such that $P(A_n)\ne0$ for every $n$, one knows that $$E(X\mid G)=\sum_nE(X\mid A_n)\,\mathbf 1_{A_n},$$ hence there exists indeed a family $(\mu_\omega)$ of probability measures indexed by $\Omega$ such that, for every $\omega$ in $\Omega$, $$E(X\mid G)(\omega)=\int_\Omega X\mathrm d\mu_\omega=\int_\Omega X(\varsigma)\mathrm d\mu_\omega(\varsigma).$$ For every $n$ and every $\omega$ in $A_n$, one defines $$\mu_\omega=P(\ \mid A_n),$$ that is, for every event $B$, $$\mu_\omega(B)=P(B\mid A_n).$$ In the general case, the assertion of your instructor is that there exists a family $(\mu_\omega)$ of probability measures indexed by $\Omega$ such that, for every $\omega$ in $\Omega$ and every event $B$, $$P(B\mid G)(\omega)=\mu_\omega(B).$$ Since the LHS is only defined almost surely, each of these identities is meant in the almost sure sense, and you can guess that "collating" all these null events for each event $B$ may become problematic. Such an object, when it exists, is called a regular conditional probability and I should probably mention that it might be slightly too advanced for the course you are following...