How can I define the integral of product spaces using characteristic functions?

[happysauce]

Homework Statement

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[itex]_{i}[/itex]u(E[itex]_{i}[/itex]). 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...

 

[gopher_p]

How do I define the integral of product spaces in terms of characteristic functions?

The same as any other measure space.

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[itex]_{i}[/itex](E[itex]_{i}[/itex]). 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 same as any other measure space.

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...

This sounds like a good strategy. But you probably want to restrict yourself to simple functions of the form [itex]f(x)=\sum a_i\chi_{E_i}(x)[/itex] where the [itex]E_i[/itex] 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?

 

[happysauce]

That's what I tried. I took the product of the integrals and expressed them as a product of two sums [[itex]\sum[/itex]a[itex]_{i}[/itex]u(E[itex]_{i}[/itex]) ][[itex]\sum[/itex]b[itex]_{j}[/itex]v(E[itex]_{j}[/itex])], 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.

 

[micromass]

That's what I tried. I took the product of the integrals and expressed them as a product of two sums [[itex]\sum[/itex]a[itex]_{i}[/itex]u(E[itex]_{i}[/itex]) ][[itex]\sum[/itex]b[itex]_{j}[/itex]v(E[itex]_{j}[/itex])], 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.

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