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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.)

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up vote 2 down vote accepted

Well, if $Z$ is constant almost surely, then certainly this constant will be $\mathbb E[Z]$.

In other words, you can choose $X=Z$ and $\mathcal A$ as the given $\sigma$-algebra.

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