Q:

Why $\Bbb{E}f$ is $\cal F_0$ measurable if $f$ is independent of $\cal F_0$?

In my professor's lecture note there is a remark saying that "$\Bbb{E}[f]$ is $\cal F_0$ measurable if $f$ is independent of $\cal F_0$". 
I think this should be easy, but I just don't see why. Can anyone help explain? Thank you!

A:

Note that $\mathbb{E}[f]$ is a constant (assuming it exists). All constant functions are measurable with respect to ANY $\sigma$-algebra. It has nothing to do with $f$ being independent of the $\sigma$-algebra.

