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In the second chapter of Radically Elementary Probability Theory [PDF], Edward Nelson gives an axiomatization of probability theory based on algebras of random variables, briefly discusses a couple of properties, and points out in passing that the law of iterated expectation holds.

How can we show this? That is, how do we get from the definition of an algebra of r.v.'s to the consequence that $ E_{\cal{B}}E_{\cal{A}} = E_{\cal{B}} $ for algebras $\cal{B} \subseteq \cal{A}$? Nelson's definition of an algebra is the usual:

By an algebra $\cal{A}$ of random variables we will always mean a subalgebra of $\cal{R}^\Omega$ containing the constants; that is, $\cal{A}$ is a set of random variables containing the constants such that whenever $x$ and $y$ are in $\cal{A}$, then $x + y$ and $xy$ are in $\cal{A}$.

The structure of an algebra $\cal{A}$ is very simple. By an atom of $\cal{A}$ we mean a maximal event [i.e., subset of $\Omega$] $A$ such that each random variable in $\cal{A}$ is constant on $A$. ...

He defines the conditional expectation $E_{\cal{A}}x(\omega)$ to be the r.v. in $\cal{A}$ that for each $\omega \in \Omega$ is the expectation of $x$ with respect to the probability relative to the atom $A_\omega$ that contains $\omega$:

$$E_{\cal{A}}x(\omega) = \frac{1}{pr(A_\omega)}\sum_{\eta \in A_\omega}{x(\eta) pr(\eta)} .$$

I imagine the law of iterated expectation follows somehow from the fact that for algebras $\cal{B} \subseteq \cal{A}$, each atom of $\cal{B}$ is the union of some number of (disjoint) atoms of $\cal{A}$, but I haven't been able to work out a complete proof. I feel like I'm missing something obvious.

(Note also that if $x \in \cal{A}$, the proof is trivial, since it is easy to show that $E_{\cal{A}}x = x $ whenever $x \in \cal{A}$. My difficulty is in showing $E_{\cal{B}}E_{\cal{A}}x = E_{\cal{B}}x$ for a r.v. $x \notin \cal{A}$.)

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This is restricted to discrete probability spaces $\Omega$, right? – Did Jul 20 '13 at 8:31
@Did, yes, everything's discrete. (The text proceeds to use non-standard analysis to extend discrete techniques to spaces that we would not be able to think of as discrete in a standard setting.) – pash Jul 20 '13 at 14:30

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