Since most posts don’t get many comments, I thought I would make one the required audience participation. The subject is “fundamental” theorems in the various subjects. What I am going for is hard to actually describe, but encapsulates a theorem being fundamental, its utility, its depth. It is the result in the subject that would hurt the most not to have, but does not have to be the putative “fundamental theorem of X”

For example, my votes for a few subjects:

Calculus: Mean Value Theorem.
Probability: Linearity of expected value.
Model Theory: The compactness theorem.

Allen Hatcher of Cornell appears to be undertaking the quixotic goal of writing accessible “introductory” textbooks for the entirety of Algebraic Topology. The first volume is already the definitive introductory work in the subject, covering the Fundamental Group, Homology, Cohomology, and Homotopy and is available online.

The style is geometric, so those whose nascent views on Algebraic Topology are functorial may be better served by Rotman’s book, but it is hard to recommend highly enough since Hatcher is an excellent expositor, and the book is clearly written for students rather than a vanity piece aimed at colleagues, a major pet peeve of mine.

Urs Lang has posted some lecture notes on geometric measure theory.

I’ve been reading Fremlin’s book, and I’ve seen a very surprising theorem that was new to me: Maharam’s Theorem. If you take an set of coins, you can define a measure space on the set of coin flips by taking the product measure. This is a probability measure: the measure of a set is the probability of a coin flip appearing in that set. Since it is a probability measure, it’s well-defined for sets of every cardinality.

You can combine any two measure spaces by taking their disjoint union; the measures are combined by addition. More generally, you can take a weighted sum. Maharam’s theorem states that every nontrivial complete measure space can be constructed from sets of coin flips by taking weighted sums. For example, counting measure is an infinite sum of flips of a single coin. Lebesgue measure on the unit interval arises from flipping an infinite number of coins.

This means that there are not very many types of (complete) measure spaces.

David Fremlin is writing the definitive tome on measure theory. He completed four volumes and is currently working on a fifth. You can find it all at the book’s web site.

For a more succinct account of measure theory, see James Hirschorn’s survey article.

A pseudodifferential operator is a generalization of a partial differential operator to fractional orders. Pseudodifferential operators allow you in some cases to invert a differential operator. For elliptic boundary value problems, they provide the easiest means to show that the surprising fact that the solution is smoother in the interior than on the boundary.

Here are some introductions to the subject:

• Essentials of pseudodifferential operators by Gabriel Nagy
• Some lecture notes on distributions and pseudodifferential operators by Gerd Grubb. The page is in Dutch, but the lecture notes are in English.
• Lectures on Pseudo-differential Operators by M. S. Joshi

I’ve tracked down some more papers on the gauge integral, also known as the generalized Riemann integral or Henstock-Kurzweil integral. The Riemann Integral Revisited offers some more details, including a proof that the characteristic function of the rationals is gauge-integrable. Non-Absolute Integrals in the Twentieth Century provides a history of the integral and some extensions.

Several standard theorems have simpler statements using the gauge integral, as shown by the following papers:

• Newton-Leibniz Formula and the Henstock-Kurzweil Integral
• Necessary and Sufficient Conditions for Differentiating Under the Integral Sign
• Stokes’ Theorem

The gauge integral is a generalization of the Riemann and Lebesgue integrals. Interestingly, there are functions that can be integrated as improper Riemann integrals but are not Lebesgue integrable. The gauge integral also subsumes improper integrals. Lebesgue integrable functions turn out to be functions f such that both f and |f| are gauge integrable.

The definition of the gauge integral is also much simpler than the Lebesgue integral, as can be seen in this presentation, An Introduction to the Gauge Integral.

Abramowitz and Stegun’s Handbook of Mathematical Functions, long the standard reference on special functions, is online. Some brave soul scanned in each page from the book (which is in the public domain).