Can I always write $Y=E[Y|X]+\varepsilon$ where $E[\varepsilon|X]=0$?

Question

Suppose that we have two random variables $Y$ and $X$. Is it true that we can always write $$ Y=E[Y|X]+\varepsilon $$ where $\varepsilon$ is some random variable such that $E[\varepsilon|X]=0$? I think this is a basic projection result but I can't seem to find it anywhere.

Answer 1

Yes, we can always do this. Simply define $\varepsilon:=Y-E(Y\mid X)$. Then calculate $$E(\varepsilon\mid X)=E(Y\mid X) - E(E(Y\mid X)\mid X)\stackrel{(*)}=E(Y\mid X) - E(Y\mid X)=0,$$ where in $(*)$ we use the fact that $E(Y\mid X)$ is already $\sigma(X)$-measurable.