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

In summary, the fundamental theorem states that if (X,B,u) and (Y,C,v) are sigma finite measure spaces and f in L1(X,B,u) and g in L1(Y,C,v), then the function h(x,y)=f(x)g(y) is also in L1(XxY,BxC,uxv). This can be proven using Fubini's theorem, and the trick to showing h is in L1 involves considering the functions hx=g(y) and hy=f(x).
  • #1
grossgermany
53
0
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:
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  • #2
Can you at least prove that h is L^1?
 
  • #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?
 

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

1. What is Fubini's Theorem?

Fubini's Theorem is a mathematical theorem that relates to the integration of multivariable functions. It states that under certain conditions, the order of integration for a double integral (or triple integral in higher dimensions) can be changed without changing the value of the integral.

2. What is the significance of Fubini's Theorem?

Fubini's Theorem is important because it allows us to simplify the process of integrating multivariable functions. Without it, we would have to use more complex methods to evaluate these integrals, which can be time-consuming and prone to errors.

3. How is Fubini's Theorem proven?

Fubini's Theorem is proven using mathematical techniques such as the Lebesgue integration and the Tonelli's theorem. The proof involves breaking down the double integral into two single integrals and then using the properties of these two techniques to show that the order of integration can be changed.

4. What are the conditions for Fubini's Theorem to hold?

There are two main conditions for Fubini's Theorem to hold: 1) the function being integrated must be continuous on the region of integration, and 2) the region of integration must be a rectangular region or a finite union of rectangles. If these conditions are met, then Fubini's Theorem can be applied.

5. How is Fubini's Theorem used in real-world applications?

Fubini's Theorem has many applications in fields such as physics, engineering, and economics. It can be used to calculate the volumes of complex shapes, find the center of mass of an object, and solve optimization problems. It is also used in probability and statistics to calculate joint probability distributions and expected values.

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