Proof of Fubini's Theorem: Integrating h(x,y)

[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

 

[morphism]

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

 

[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?