I am working on a problem in probability theory and would appreciate your help. The problem is stated as follows:
Given a measure space $(\Omega, \mathcal{F}, P)$, a random variable $X : \Omega \to \mathbb{R}$ and a $\sigma$-algebra $\mathcal{A} \subseteq \mathcal{F}$, suppose that for every $A \in \mathcal{A}$, we have
$$\int_A \left( \mathbb{E}(X|\mathcal{A}) - \mathbb{E}(X) \right) dP = 0.$$
Does this imply that $X$ is independent of the $\sigma$-algebra $\mathcal{A}$? If it does, could you provide a proof? If it does not, could you provide a counterexample?
My initial thoughts:
The given condition indicates that the deviation of the conditional expectation of $X$ given $\mathcal{A}$ from the overall expectation of $X$ integrates to zero over all $A \in \mathcal{A}$. It isn't immediately clear how this condition relates to the concept of independence between a random variable and a $\sigma$-algebra. Independence would mean that knowing which event in $\mathcal{A}$ occurred does not provide any information about $X$, i.e., the conditional distribution of $X$ given any event in $\mathcal{A}$ is the same as the unconditional distribution of $X$. However, the given condition does not seem to directly provide information about the distribution of $X$ given events in $\mathcal{A}$, so I am not sure how to make the connection to independence.
I would greatly appreciate any insights you might have into this problem. Thank you!