Consider conditional expectation of a real-valued r.v. $X$ given a sub sigma algebra $\mathcal{B}$ of the probability space $(\Omega, \mathcal{F}, P)$.
Will independence between $\sigma(X)$, i.e. the sigma algebra of the r.v., and the given sub sigma algebra $\mathcal{B}$ make $E(X \mid \mathcal{B}) \equiv E(X)$ ?
Is independence between $\sigma(X)$ and the given sub sigma algebra $\mathcal{B}$ is the only way to define independence between $X$ and $\mathcal{B}$?
If there are other ways, will they make $E(X \mid \mathcal{B}) \equiv E(X)$ ?
Consider conditional expectation of a real-valued r.v. $X$ given another $S$-valued r.v. $Y$ on the same probability space $(\Omega, \mathcal{F}, P)$.
Will independence between $X$ and $Y$ make $E(X \mid Y) \equiv E(X)$ ?
Why? References are also appreciated! Thanks and regards!