Can I conclude that $E[X\mid \mathcal{F}]=E[X]$ implies $X$ is independent of $\mathcal{F}$.I know that $X$ is normally distributed. (This would ofocurse on work if X is a constant since very constant is trivially $\mathcal{F}$ measurable)
Applying the definition of conditional expectation I get for any $A \in \mathcal{F}$
$E[X 1_A]=E[E[X]1_A]=E[X]E[1_A]$
Now if $X$ and $\mathcal{F}$ are independent we can claim that the product of expectations is the expectation of the product. But how to we prove independence in this case. It appears to me that this condition doesnt imply independence. But I cant think of a counterexample. Can somebody provide me a counterexample?
If $X$ has standard normal distribution then $EX=0=E(X|X^{2})$, but $\{X,X^{2}\}$ is not independent.