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Can someone please share references (websites or books) where I can find problems related with Fubini's theorem and applications of Radon-Nikodym theorem? I have googled yes and don't find many problems. What are the "typical" problems (if there are any) related with these topics? [Yes, exam is coming soon so I don't know what to expect and don't have access to midterms from previous years].

Thank you

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3 Answers 3

One basic example of what you can do with Fubini's theorem is to represent an integral of a function of a function in terms of its distribution function. For instance, there is the formula (for reasonable $\phi$, $f$ basically arbitrary but nonnegative) $$ \int \phi \circ f d\mu = \int_t \mu(\{ f> t\}) \phi'(t) dt$$ which reduces questions about integrals of, for instance, $p$th powers to questions about the distribution function. This is (for example) how the boundedness of the Hardy-Littlewood maximal operator on $L^p$ ($p>1$) is proved: you get a bound on the distribution function of the maximal function by general methods and then do an interpolation process.

To prove the formula above, as in Rudin, one can consider the collection $E$ of pairs $(x,t)$ such that $0 \leq t \leq f(x)$. This is a measurable subset of $X \times \mathbb{R}$ if $X$ is the initial measure space on which $f$ is defined and $\mathbb{R}$ has Lebesgue measure. Then, one can write the second integral in the displayed equation as $\int_t \phi'(t) dt \int_{x \in X} \chi_E(x,t) d \mu$ where $\chi_E$ denotes the characteristic function.

Now rearranging this integral via Fubini's theorem allows one to integrate with respect to $t$ first, for each $x$; then $t$ goes from $0$ to $f(x)$, and one can see that this integral becomes $\int_x \int_{t=0}^{f(x)} \phi'(t) dt$, which is the right-hand-side of the displayed equation.

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I would like to add that the theory around the Hardy-Littlewood maximal function is very interesting and nicely explained in Stein's book "Singular integrals and differentiability properties of functions". –  Jonas Teuwen Jan 6 '11 at 22:15

Radon-Nikodym is used to prove the existence of the conditional expectation in probability theory.

Fubini's theorem is, among other things, a very useful device to compute integrals over product spaces.

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Actually, Radon-Nikodym is just used to show the existence of the conditional expectation. It's non-constructive. =) –  Jens Nov 30 '10 at 7:44
The proofs of Radon-Nikodym I have seen are reasonably constructive. To find $f = d\nu/d\mu$, where $\mu, \nu$ are finite positive measures, you consider the collection of all functions $g$ such that $\int_A g d\mu \le \nu(A)$ for all measurable sets $A$, and then take $f$ to be the supremum of all such $g$. Then one just has to verify that $f$ has the desired properties of the Radon-Nikodym derivative. The versions for $\sigma$-finite or complex measures are easy extensions. –  Nate Eldredge Nov 30 '10 at 16:22
There is also a nice martingale proof (with the martingale convergence theorem). –  Jonas Teuwen Nov 30 '10 at 17:17

I am not expert, but my own limited view is this. I said for thirty years I could see no interest in theorems like radon nikodym, having learned them very abstractly. then once while teaching honors calculus i asked myself what the fundamental theorem of calculus should say for integrals of just Riemann integrable, not necessarily continuous functions.

After originally stating the wrong answer to my class, I learned (with help from an analyst friend who showed me a Cantor function) that the indefinite integral is characterized by being a function which has a derivative equal to the original integrand almost everywhere (wherever that integrand is continuous) and being also not just continuous but Lipschitz continuous.

Then I realized at last the radon nikodym theorem is just the fundamental theorem of calculus for more general functions. We may not realize the analogy from calc 1, because by considering only continuous integrands there we miss out on the singular part. I.e. we forget to wonder why we are looking only at integrals of derivatives of C^1 (in particular Lipschitz) functions. So any application of FTC type is an application of RN, e.g. it lets you characterize constant functions (a.e?) by weak continuity and differentiability properties.

Similarly, Fubini is of course repeated integration, so reduces any integral computation inductively to one of lower dimension (volumes by slicing). E.g. to show some set has measure zero (as in Sard's theorem) you can do it inductively by showing most of the slices have lower dimensional measure zero. (See Guillemin and Pollack, appendix, or Milnor's differential topology book.)

One of my professors once suggested that virtually all problems in analysis are attacked by either dominated convergence or Fubini. So if you see an exam problem that dominated convergence won't do, try Fubini.

This answer assumes you retain an interest in the topic even after your exam, unless you are at Harvard, where exams are perhaps still in january.

Following up KCd's comment, you might also peruse old qualifying exams available on the websites at schools like Harvard and UGA. Harvard's also has a few lists of typical questions of this nature: e.g. if every intersection of a certain subset S of the plane, with a line of slope 1 is countable, what can you say about the Lebesgue measure of S? Gosh, Harvard even has, with login capability, online copies of all exams since 1977, and paper copies in libraries of exams since 1836!

Or consider a continuous weakly monotone increasing function f on the interval [0,1]. One knows that f is differentiable a.e. say with derivative g ≥ 0. If g is integrable and G(x) is the integral of g from 0 to x, then to what extent does G determine f or g, if either? When does G determine both f and g? Give an example if possible where G does not determine f, respectively g.

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This is not very important, but Harvard has exams in December. –  Akhil Mathew Dec 31 '10 at 0:51
They had final exams there in January until only the last couple of years. (Moreover, I assumed Roy was talking about the graduate qualifying exams, which are still given in January.) –  KCd Dec 31 '10 at 22:23
Well, I confess I was remembering suffering through xmas vacations, unable to relax or study, in 1960. I'm glad things have evolved, at least for undergrads. As I recall, at Harvard these advanced grad courses were also loaded with young kids from the College. I think even I foolishly started fall semester freshman year in math 280, at least for a while, and I was a public school student. Top undergrad majors took 212 (measure theory) as juniors. –  roy smith Jan 5 '11 at 20:18

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