# Measurability when conditioning on a sub-sigma field

I'm currently going through a book on measure-theoretic probability. In it, we defined conditional expectation:

Let X belong to $$M^+(Omega, F)$$ and P be a probability measure on F. For each sub-sigma-field G of F, the conditional expectation$$P(X|G)$$ is the random variable $$X_G$$ in $$M^+(X,G)$$ for which $$P(gX) = P(gX_G)$$.

However, there is a small disclaimer: be careful to check that $$X_G$$ is G-measurable. I'm confused about how this can be a problem, since isn't $$X_G\in M^+(X,G)$$? Doesn't this by definition make it $$G-$$measurable?

• \Omega produces $\Omega$ – parsiad Apr 3 '19 at 0:40

They are just saying that if you want to prove that $$Y$$ is $$P(X|G)$$ you have to verify two facts: $$Y \in M^{+}(X,G)$$ and the condition $$P(gX)=P(gY)$$. A common mistake it to ignore the first part.