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Product measure

  1. Dec 5, 2008 #1
    What's the proof of this fundamental theorem?
    Let (X,B,u) and (Y,C,v) be sigma finite measure spaces, Let f in L1(X,B,u) and g in L1(Y,C,v). Let h(x,y)=f(x)g(y).
    Then, h is in L1(XxY,BxC,uxv) and
    [tex]\int hd(u\times v)=\int fdu \int gdv[/tex]

    should be an easy application of fubini,but i really have no idea to how work it out
     
    Last edited: Dec 5, 2008
  2. jcsd
  3. Dec 5, 2008 #2

    morphism

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    Can you at least prove that h is L^1?
     
  4. Dec 5, 2008 #3
    i know the trick must has something to do with
    hx=g(y)
    hy=f(x)
    but then h=hx*hy, why is it in L1?
     
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