I have a very basic question concerning conditional expectation.
Let $(\Omega, \Sigma, p)$ be a (finite) probability space and let $\Sigma_k$ be a sub-$\sigma$-algebra (with $k \in K := \{1,2\}$), where $\Sigma_1$ and $\Sigma_2$ are not necessarily equal.
Also, let $X : \Omega \to \mathbb{R}$ be a random variable and let $A \in \Sigma$ such that for every $x \in A$ there is a $S_x \in \Sigma_k$, for every $k \in K$, such that $S_x \subseteq A$. Finally, let $\Sigma^{A}_k$ be the corresponding partition of $A$ induced by $\Sigma_k$.
Here there is the question:
1) Can we say that $E [ X | \Sigma^{A}_1] = E [ X | \Sigma^{A}_2]$?
(Alternatively, is it possible to have $E [ X | \Sigma^{A}_1] > 0$ and $E [ X | \Sigma^{A}_2] < 0$?)2) How does this work when the cardinality of $K$ is larger than $2$?
I would say yes to the first question (and no to its reformulation, since we are basically conditioning on $A$. Moreover, I would say that nothing changes when the cardinality of $K$ gets larger, but how to proceed to prove it?
Any feedback will be most welcome.