Proving Fubini's Theorem: Tips & Ideas

[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?

Thanks for your help.

 

[quasar987]

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

 

[Hurkyl]

Well, you can always try proving it yourself.

 

[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!

 

[amcavoy]

Well, you can always try proving it yourself.

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.

 

[Hurkyl]

It ought to be easier to prove

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

than to prove

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

 

[amcavoy]

It ought to be easier to prove

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

than to prove

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

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

 

[Hurkyl]

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?

 

[quasar987]

I tought Fubini's theorem was just

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

 

[Hurkyl]

With some conditions on f.

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

 

[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?

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

 

[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)

 

[amcavoy]

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)

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

 

[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.

 

[amcavoy]

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.

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

 

[quasar987]

Yes I know it (worked through it on PF a few months back!)

Could you post the link please? I can't find the thread.

As for the statement, Castilla was probably referring to

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

 

[amcavoy]

Could you post the link please? I can't find the thread.

As for the statement, Castilla was probably referring to

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

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

https://www.physicsforums.com/showthread.php?t=36033

 

[Castilla]

I tought Fubini's theorem was just

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

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

 

[amcavoy]

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

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.

 

[quasar987]

I think Castilla is saying that fubini is a corolary, or at least that the proof is similar, to Leibniz rule (the theorem stated at proved by homology in post #15: https://www.physicsforums.com/showpost.php?p=264381&postcount=15)

 

[amcavoy]

I think Castilla is saying that fubini is a corolary, or at least that the proof is similar, to Leibniz rule (the theorem stated at proved by homology in post #15: https://www.physicsforums.com/showpost.php?p=264381&postcount=15)

Thanks quasar I hadn't seen that post before. That makes sense then how to prove Fubini's Theorem in a similar manner. I'm going to try writing it out and see what I get.

Thanks again for the help

 

[Castilla]

Fact: the function with domain $$[a,b]x [c,d]$$ is continuous.

Let be $$g(x,z) = \int_c^zf(x,y)dy.$$ Then, by the Fundamental Theorem of Calculus, $$\frac{\partial g}{\partial z}= f.$$
Let be
$$G(z) = \int_a^bg(x,z)dx$$, so by differentiation under the integral sign
$$G'(z) = \int_a^bf(x,z)dx$$.

Assume that G' is continuous. Then

$$\int_c^d\int_a^bf(x,z)dxdz = \int_c^dG' = G(d) - G(c) = G(d) =<br /> \int_a^b\int_c^df(x,y)dydx$$.

 

[amcavoy]

Fact: the function with domain $$[a,b]x [c,d]$$ is continuous.

Let be $$g(x,z) = \int_c^zf(x,y)dy.$$ Then, by the Fundamental Theorem of Calculus, $$\frac{\partial g}{\partial z}= f.$$
Let be
$$G(z) = \int_a^bg(x,z)dx$$, so by differentiation under the integral sign
$$G'(z) = \int_a^bf(x,z)dx$$.

Assume that G' is continuous. Then

$$\int_c^d\int_a^bf(x,z)dxdz = \int_c^dG' = G(d) - G(c) = G(d) =<br /> \int_a^b\int_c^df(x,y)dydx$$.

Thank you very much Castilla. That wasn't quite what I was doing, so I must have been doing it incorrectly (or going off track).

Thanks again.

 

[quasar987]

Pretty g.d. sweet. But I wouldn't have found that alone.

 

[Hurkyl]

Incidntally, it can be proven under other conditions (which is why I asked about the statement of the theorem):

For example, when f is bounded, discontinuous on a set of measure 0, and any horizontal or vertical line passes through only finitely many points of discontinuity.

 

[Castilla]

I think that the version developed is known as "Baby Fubini's theorem". There is a second version which includes double integrals (not only iterated, as the "Baby"), and Hurkyl refers to a third more "mature" version.

 

[quasar987]

I think that the version developed is known as "Baby Fubini's theorem". There is a second version which includes double integrals (not only iterated, as the "Baby"), and Hurkyl refers to a third more "mature" version.

What do you mean by "double integrals, not only iterated"?

 

[Castilla]

1. Iterated integrals: $$\int_a^b(\int_c^dg(x,y)dx)dy$$ or $$\int_c^d(\int_a^bg(x,y)dy)dx$$. The "Baby Fubini T." only says that (under certain conditions) these are equal.

2. A double integral:$$\int\int_Rg(x,y)dxdy$$ where R = (a,b) x (c,d). The "not so baby" Fubini T. says that this double integral equals the iterated integrals of the previous paragraph.

(Of course this can be extended to three iterations, etc.).

But Hurkyl's one is beyond these two statements (i believe).

 

[Hurkyl]

Accoding to my calc III book, Fubini proved a "very general version of this theorem", but the case of continuous f was known to Cauchy a century beforehand.