I know that the following is true and fairly easily proven.
Let $Y$ be a random variable and $\varphi$ a measurable function.
Let $A$ be a $\Sigma_Y$ measurable set.
If $ X (\omega) = \varphi(Y (\omega))$ for all $\omega\in A $ , then $\mathbb{E}(X|Y )(\omega) = \varphi(Y (\omega))$ for almost all $\omega\in A $
I cannot, however, come up with an example of a variable $Y$ and function $\varphi$ such that $ X (\omega) = \varphi(Y (\omega))$ for almost all $\omega\in A $ (and $A$ is not $\sigma(Y)$ measurable) but $\mathbb{E}(X|Y )(\omega) = \varphi(Y (\omega))$ .
Could you help me with that?
EDIT: I'm looking for $Y$ such that for $A \not\in \sigma(Y)$, $\omega \in A$ $$ X (\omega) = \varphi(Y (\omega))$$ but $$\mathbb{E}(X|Y )(\omega) \neq \varphi(Y (\omega))$$