I just came across something that I didn't notice before about conditional expectation - Is this correct:
If $Z = \text{const. a.s.}$, then $Z=\mathbb{E}[X]$ a.s.,
where $Z = \mathbb{E}[X \mid \mathcal{A}]$ for a random variable $X$ and a $\sigma$-algebra \mathcal{A}?
(I would conclude this from the property \mathbb{E}[ X 1_A] = \mathbb{E}[Z 1_A]$, where for $Z(\omega) = z$ a.s., $A=\Omega$ I get $\mathbb{E}[X] = z$ a.s.)