Fubini's Theorem

1. Oct 1, 2005

amcavoy

Does anyone know how to prove Fubini's Theorem? In multivariable calc., I just accepted it as true but never learned the formal proof. Any ideas?

2. Oct 1, 2005

quasar987

Same here. Everywhere I turn to, authors seem to flee this proof like the plague.

3. Oct 1, 2005

Hurkyl

Staff Emeritus
Well, you can always try proving it yourself.

Last edited: Oct 1, 2005
4. Oct 1, 2005

quasar987

I did. Tought I had it but turns out I had made a limit switching without realizing it. I don't know if the limit switching is valid, but it probably isn't, otherwise the proof is just so direct, authors wouldn't say its "above the level of this course".

Actually it probably is valid, but it's proving its validity that's very hard!

5. Oct 1, 2005

amcavoy

I'd like your opinion on this: How does one go about justifying switching limits? Could you maybe post a simple example that I could try applying to this?

I appreciate it.

6. Oct 1, 2005

Hurkyl

Staff Emeritus
It ought to be easier to prove

$$\iint_R f = \int_a^b \int_c^d f(x, y) \, dy \, dx$$

than to prove

$$\int_a^b \int_c^d f(x, y) \, dy \, dx = \int_c^d \int_a^b f(x, y) \, dx \, dy$$

7. Oct 1, 2005

amcavoy

So basically it comes down to writing down the definition of an integral (limit of a series) and rearranging, right?

8. Oct 1, 2005

Hurkyl

Staff Emeritus
Well, it mainly involves doing a lot of putting bounds on things (just like any other "interesting" proof in analysis). By the way, what statement of Fubini's theorem are you trying to prove?

9. Oct 1, 2005

quasar987

I tought Fubini's theorem was just

$$\int_a^b \int_c^d f(x, y) \, dy \, dx = \int_c^d \int_a^b f(x, y) \, dx \, dy$$

10. Oct 1, 2005

Hurkyl

Staff Emeritus
With some conditions on f. :tongue2:

(And that both of those iterated integrals are equal to the double integral)

11. Oct 1, 2005

amcavoy

Hmm... Maybe it's time for something similar, but a bit simpler. Is the proof for the following in differential calc. similar to the proof of Fubini's Theorem?

$$\frac{\partial^2\,f}{\partial x\,\partial y}=\frac{\partial^2\,f}{\partial y\,\partial x}$$

12. Oct 1, 2005

quasar987

For your personal amusement apmcavoy, this thm of diff. cal. is called Clairaut's Theorem (also sometimes called Schwartz's Theorem...but Schwartz already has his inequality, so let's be fair and give this one to Clairaut, shall we :P)

13. Oct 1, 2005

amcavoy

lol. I am finding that my multivariable class was more and more useless as I encounter these theorems we never proved. Ahh, I hope these aren't expected until analysis

14. Oct 2, 2005

Castilla

Apmcavoy, do you know the proof of the Leibniz rule (differentiation under the integral sign)? From said proof is rather easy to obtain the statement wrote by Quasar.

Put "Leibniz rule" in google and you will see it.

Castilla.

15. Oct 2, 2005

amcavoy

Yes I know it (worked through it on PF a few months back!) Which statement made by Quasar are you referring to?

16. Oct 2, 2005

quasar987

As for the statement, Castilla was probably refering to

$$\int_a^b \int_c^d f(x, y) \, dy \, dx= \int_c^d \int_a^b f(x, y) \, dx \, dy$$

17. Oct 2, 2005

amcavoy

There are a few, but this was the one I could find:

18. Oct 2, 2005

Castilla

This equation. If this is what you want to proof, it can be considered a corolary from Leibniz rule (differentiation under the integral sign).

19. Oct 2, 2005

amcavoy

Hmm... Integration under the integral sign? You know what the thread on the Leibniz rule never explained: justification for switching limits. I'd like to know how to do this. I mean, I know that you can if the limits of integration are constants and the function is continuous, but that isn't really a proof.

20. Oct 2, 2005