Let $V$ be the set of all (real-valued) random variables defined on the probability space $(\Omega,\mathcal F,\Bbb P)$. It is easy to verify that $V$ is a real vector space.

Question: What is an example of a real basis of $V$?

A natural guess would be the set of indicator random variables $\{1_A\mid A\in\mathcal F\}$, but I am unable to prove that it is either linearly independent or spanning.

  • $\begingroup$ Indicators are not linearly independent, e.g. $1_{A \cup B} = 1_A + 1_B - 1_{A \cap B}$. $\endgroup$ – Robert Israel Mar 2 '17 at 2:14

Let's suppose $\mathcal F$ is finite (I doubt that you'll get an explicit basis in a case where $\mathcal F$ is infinite). Then there is a finite collection of atoms $A_i$, $i=1..m$, which are minimal nonempty members of $\mathcal F$. The indicators of these atoms form a basis.

  • 1
    $\begingroup$ Thanks. Linear independence is straightforward, and the fact that they span is because random variables are constant on atoms and that each $\omega\in\Omega$ is contained in some atom. Is the assumption that $\mathcal F$ is finite made to ensure that $\mathcal F$ is atomic? $\endgroup$ – Empty Set Mar 2 '17 at 4:00

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