2
$\begingroup$

Let $X$ be a random variable on a probability space $(\Omega, \mathcal A, P)$ such that $E|X|<\infty$. Suppose $\mathcal G \subset \mathcal A$ is a sub-$\sigma$-algebra with intersection-stable generator $\mathcal M$ such that $\Omega \in \mathcal M$. Let $Y\in L^1(\Omega, \mathcal G, P)$. Then $$\int_AX dP =\int_AY dP$$ for every $A\in \mathcal M$ implies that the above equality holds for every $A \in \mathcal G$.

How do I approach this problem? The first thing that jumps to mind whenever we have an intersection-stable generator is that its generated Dynkin-system coincides with its generated $\sigma$-algebra. Does this imply that we can write $A\in \mathcal G$ as the disjoint union of sets that lie in $\mathcal M$ or whose complement lies in $\mathcal M$?

I'm thankful for any help.

$\endgroup$
  • $\begingroup$ Related? $\endgroup$ – BCLC Sep 18 '15 at 13:23
2
$\begingroup$

If $\mathcal D:=\{A\in\mathcal G\mid \int_AXdP=\int_AYdP\}$ then it can be proved that $\mathcal D$ is a Dynkin-system.

This with $\mathcal M\subseteq\mathcal D$ so that $\mathcal D$ will contain the Dynkin-system generated by $\mathcal M$.

Since $\mathcal M$ is intersection-stable this collection will coincide with the $\sigma$-algebra generated by $\mathcal M$ wich is $\mathcal G$ (as you said).

So $\mathcal G\subseteq\mathcal D$ as was to be shown.

$\endgroup$
  • $\begingroup$ Related? $\endgroup$ – BCLC Sep 18 '15 at 13:23
  • 1
    $\begingroup$ @BCLC Yes, there is an evident relation. $\endgroup$ – drhab Sep 18 '15 at 15:56
  • $\begingroup$ does this property hold for non sigma-finite measures? (i.e., if we replace $P$ by an arbitrary measure $\mu$) $\endgroup$ – user2139 May 3 '18 at 6:43
  • 1
    $\begingroup$ @user2139 Yes. For integrable functions $X,Y$. $\endgroup$ – drhab May 3 '18 at 7:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.