When are the limit operations commutative?

I'll come up with a question that has bothered me for a long period of time. The question seems relatively simple, but I personally didn't manage to find an answer to it. In many cases I met problems where I had to deal with multiple limits and I've never known what is the rule that allows the limits to be interchanged. I'm referring at the situation when the limit operations are commutative. For some problems this would be of enormous help since the problem you have to deal with becomes much more easier when you change the limit operations. Could you help here? Thanks.

Solution 1:

This question is not trivial to answer with any finality. The "crisis" in 19th century analysis could be said to hinge on such issues, and it took many very capable people many decades to even develop language to frame the question precisely!

Indeed, in real life situations, one must hope to be able to interchange various limits to proceed further in many problems. The question is not whether we want to or not (we do), but whether it is ok, and/or under what additional hypotheses it might be ok, ... and how to prove these things. Even in situations where (with hindsight) there is no genuine danger at all in interchanging limits, proving this can be non-trivial, if only because the vocabulary needed for the proofs is typically more sophisticated than the context otherwise.

rschweib's excellent example of Fubini/Tonelli is a very accessible example, because the two limits (integrals) are obviously "of the same sort" and have a physical sense. Further, even a mildly theoretical calculus preparation gives one an idea of what is involved.

A trickier important class of examples is about differentiation with respect to a parameter under the integral sign, as in ${\partial\over \partial t}\int_a^b F(x,t)\,dx$, where $F$ is differentiable in $t$, and perhaps continuous in $x$, and there are potential worries about the limits of integration being at $\pm \infty$, or weakening hypotheses on $F$ as a function of $x$. Various "advanced calculus" or "introductory analysis" books have theorems giving sufficient hypotheses for differentiating under the integral, but I confess I have never found these results satisfying or memorable. Rather, a somewhat more sophisticated viewpoint (but almost a century old) is to think of function-valued functions, and metrics or topologies on these spaces, in which the operations of integration and differentiation are continuous. So $F(x,t)$ might be viewed as a continuous-function-of-$x$-valued differentiable function of $t$. Integration-in-$x$ should/could be a continuous linear map from those function-valued-functions to scalar-valued functions, and differentiation in $t$ should/could be a continuous linear map from differentiable function-valued-functions to continuous function-valued-functions. (I fear that this is a fancier kind of explanation than desired, but it is a viewpoint that finally gave me a feeling of confidence in proofs of these things, rather than merely "hope"...) So, then, our task would be to make these sketched ideas precise... and then, often, the desired conclusion is immediate, just based on understanding the set-up.

A notorious historical example of lack of vocabulary eventually evolving to a more sophisticated situation that allowed clear proofs is the "Dirichlet principle", invoked by Riemann and many others, but which was shown by Weierstrass to need further clarification, to say the least. Around 1900, Hilbert, B. Levi, and Fubini systematically addressed such issues, and then Bochner, Weyl, Sobolev, L. Schwartz, Grothendieck, and many others very nicely clarified ideas about _spaces_of_functions_ and function-valued functions. Some cost, but well worth it. Not accessible to beginners in calculus, no, but, given the history of the confusions, this is entirely fair! :)

Solution 2:

I think an interesting related phenomenon you'll want to see eventually is switching the order of integration. Integration is, after all, a limiting process. Moreover, switching the order of integration is often the way to reach a solution!

Fubini's theorem is an example theorem that gives conditions when the order of integration can be switched. I remember while doing entry Exams on real analysis, there was a really nice example given of a function whose integral was dependent on order. (In fact, it got different answers for each of the two iterated integrals, and a third completely different answer using the product space integral.) I will try to remember to write it in later today!