0
$\begingroup$

I am learning about product measures and I was stuck on a detail of the proof. I would appreciate any assistance!

Suppose we have a measure spaces $(X_i, M_i, \mu_i), i=1, ..., N$, that are complete and $\sigma$-finite. We call sets measurable rectangles, if it has the following form $$ A_1 \times ... \times A_N, $$ where each $A_i \in M_i$.

Let $M_0$ denote the collection of sets that are finite disjoint union of measurable rectangles. Then $M_0$ turns out to be an algebra of subsets of $X_1 \times ... \times X_N$. I am having hard time seeing how $M_0$ is an algebra. Could someone please explain this to me? Thank you!

$\endgroup$
0
$\begingroup$

Well, you have to prove that it is closed in respect to finite union, intersection and complement. The union is pretty clear (it is the definition of $M_0$), the complement doesn't seem to be very difficult too... As for the intersection, I think that you should first analyse an intersection of two rectangles. And then you decompose the intersection of a union of such rectangles into the union of intersections.

$\endgroup$

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.