Integral of product spaces

1. Dec 2, 2012

happysauce

1. The problem statement, all variables and given/known data

I just have a question about the integral of a product space. How do I define the integral of product spaces in terms of characteristic functions?

What I mean by that is, if I have a measure space, (X,M,u) and f(x) is a positive, simple, measurable function. Then ∫f du = Ʃa$_{i}$u(E$_{i}$). What I want to know is how can I apply this to a simple function given the product space of (X,M,u) and (Y,N,v)?

The problem I have to do is to prove that ∫f(x)g(y)d(u×v)=(∫f(x)du)(∫g(y)dv), you can't use fubini's theorem since the problem doesn't assume the measure spaces are sigma finite and the hint suggested using standard limit theorems for integrals, which made me think I probably had to use simple functions...

2. Dec 2, 2012

gopher_p

The same as any other measure space.

The same as any other measure space.

This sounds like a good strategy. But you probably want to restrict yourself to simple functions of the form $f(x)=\sum a_i\chi_{E_i}(x)$ where the $E_i$ are "rectangles" in the product space. Can you decompose the characteristic function of a rectangular region into a product of characteristic functions on the "component" spaces?

3. Dec 3, 2012

happysauce

That's what I tried. I took the product of the integrals and expressed them as a product of two sums [$\sum$a$_{i}$u(E$_{i}$) ][$\sum$b$_{j}$v(E$_{j}$)], one summed n parts the other summed m parts. Then I noticed that the product created a sum of n×m rectangles and I was able to express it in terms of one sum.

4. Dec 3, 2012

micromass

Staff Emeritus
OK, so you proved the theorem for the special case simple functions, where for every $\chi_{E_j}$ holds that $E_j$ is a rectangle. What did you do next?