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I know that the following is true and I've already proven it.
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 counterexample which proves that the above statement can be false if $A \not\in \Sigma_Y$.
Could you help me with that?
That is, I'm looking for a random variable $Y$ sych that for $A \not\in \sigma(Y)$ , $\omega \in A$ $$ X (\omega) = \varphi(Y (\omega))$$ but $$\mathbb{E}(X|Y )(\omega) \neq \varphi(Y (\omega)).$$
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 counterexample which proves that the above statement can be false if $A \not\in \Sigma_Y$.
Could you help me with that?
That is, I'm looking for a random variable $Y$ sych that for $A \not\in \sigma(Y)$ , $\omega \in A$ $$ X (\omega) = \varphi(Y (\omega))$$ but $$\mathbb{E}(X|Y )(\omega) \neq \varphi(Y (\omega)).$$