1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integral of product spaces

  1. Dec 2, 2012 #1
    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[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...
     
  2. jcsd
  3. Dec 2, 2012 #2
    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 [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?
     
  4. Dec 3, 2012 #3
    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.
     
  5. Dec 3, 2012 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Integral of product spaces
  1. Product Spaces (Replies: 6)

  2. Integral of product (Replies: 3)

  3. Integral of products? (Replies: 2)

Loading...