Product measure

1. Dec 5, 2008

grossgermany

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
$$\int hd(u\times v)=\int fdu \int gdv$$

should be an easy application of fubini,but i really have no idea to how work it out

Last edited: Dec 5, 2008
2. Dec 5, 2008

morphism

Can you at least prove that h is L^1?

3. Dec 5, 2008

grossgermany

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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