[Math] Conditional expectation and independence

Question

1. 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 $E(X \mid \mathcal{B}) \equiv
   E(X)$ ?

2. 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!

Answer 1

The answer to the first question in 1. is yes.

As a special case, the answer to 2. is also yes.

Concerning the second question in 1.: It's the only way I can think of. One definition is enough, isn't it? You can of course spell out what indepence actaully means in this case and take what you get as your definition.

Concerning the third question in 1.: They should, otherwise they cannot be equivalent defitions.