Expectation conditioned on a sub sigma field Let $X$ and $Y$ be two integrable random variables defined on the same probability space $(\Omega,\mathcal F,\mathbf P)$
Let $\mathcal A$ be a sub-sigma-field such that X is $\mathcal A$-measureable.


*

*Show that $E(Y|A)=X$ implies $E(Y|X)=X$

*Show by counter-example that $E(Y|X)=X$ does not imply that $E(Y|A)=X$


Because $X$ is $A-$measurable, $\sigma(X)\subseteq A$ therefore:  
$$
\mathbb{E}(Y|X)=\mathbb{E}[\mathbb{E}(Y|A)|X]=\mathbb{E}[X|X]=X.
$$
For the next part, suppose that $\sigma(X)\subset A$ therefor there is a set $G\notin \sigma(X)$ and $G\in A$ and we have $\mathbb{E}[\mathbb{1}_G|X]=\mathbb{P}(G|X)=p>0$. Consider $Y=\frac{1}{p}X\mathbb{1}_G$. Then $\mathbb{E}[Y|X]=X$ however $\mathbb{E}[Y|A]=\frac{X}{p}\mathbb{E}[\mathbb{1}_G|A]=\frac{1}{p}X\mathbb{1}_G$ which is not equal to $X$. 
