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I am having some trouble understanding how to apply Fubini and/or Tonelli Theorems to determine whether a Lebesgue integral over $\mathbb{R}^2_+$ exists and if it is finite.

If someone could help me by showing the explicit steps for the examples below I would be grateful. I have a long list of exercises I have found online (this is self-study) and instead of posting a bunch of examples here I thought a few simple ones would help me learn how to do these problems going forward.

The examples in hand are: for each of the functions, use Fubini or Tonelli to show the existence/finiteness of the function's Lebesgue integral over $\mathbb{R}^2_+$.

$f_1(x,y)=\frac{\sin xy}{xy}$

$f_2(x,y)=e^{-(1+x^2)y}$

Many thanks in advance!

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From FAQ about tags: Try to avoid creating new tags. Instead, check if there is some synonym that already has a popular tag. It's not easy to keep balance between too specific tags and not having enough tags, but tags for specific theorems are usually too specific, perhaps with the exception of some very important ones. (Of course, you can disagree, and there is possibility for further discussion, if needed.) –  Martin Sleziak Aug 16 '12 at 12:58
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1 Answer

up vote 2 down vote accepted
  1. Assume $f_1$ is integrable in $\mathbb{R}^2_+$ (which I assume is $\mathbb{R}\times \mathbb{R}_+$), by Fubini's theorem $f_1^x(y)=f_1(x,y)$ would be (Lebesgue!) integrable for almost every $x\in \mathbb{R}$, but $$ \int_{\mathbb{R}_+} f_1^x(y) dy= \int_0^{\infty} \frac{\sin(xy)}{xy} dy $$ and this last is not a Lebesgue integral for $x\neq 0$ (the positive and negative parts of the function $\sin(z)/z$, when integrated give $\infty$). So $f_1$ is not integrable in $\mathbb{R}^2_+$.

  2. Since $f_2$ is positive everywhere, Tonelli's theorem guarantees that we can integrate first over $y$ and then over $x$ to get $$ \int_{\mathbb{R}^2_+} f_2(x,y)d(x,y) = \int_{\mathbb{R}} \int_0^\infty e^{-(1+x^2)y}dydx= \int_{-\infty}^{\infty} \frac{1}{1+x^2} dx <\infty $$

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I think you mixed Fubini theorem and (Fubini-)Tonelli theorem in part 2). –  D. Thomine Aug 11 '12 at 23:29
    
The way I learned it was that Fubini needs integrability (or non-negativity, which is what we have in the first part of 2.) in the product space and gives the equality of the iterated integrals, and Tonelli needs finiteness of one iterated integral (of the absolute value, which is what gives the calculation in 2.) and gives integrability in the product space. With this I think I used them correctly... –  Jose27 Aug 11 '12 at 23:34
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From what I learned (which coincides with Wikipedia and the notes of T. Tao on measure theory, among others): Fubini deals with integrable functions, Tonelli with non-negative functions. –  D. Thomine Aug 12 '12 at 0:01
    
Yes, Folland and Dudley agree with you, I'll edit the answer. As an aside I still think my way of remembering them makes more sense: One goes up and the other goes down, regardless of signs. –  Jose27 Aug 12 '12 at 0:13
    
The exercise admits that an infinite integral can still be considered to "exist". Does this confuse the answer above in any way, as it relates to #1? –  Justin Aug 13 '12 at 15:21
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