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).

A new issue of the Bulletin of the AMS, Volume 32, Number 3, has hit the virtual newstands.

Charles Weibel, who wrote the best book on homological algebra is now working on a book on algebraic K-theory. He has posted drafts of the first four chapters at An introduction to algebraic K-theory.

Well, okay, not really lies, but I formed ideas in my abstract algebra class that I later had to unlearn:

• Most integral domains are unique factorization domains. In reality, integral domains are almost never UFDs. Beyond the examples usually taught, there is one additional large class of UFDs, regular local rings, and then just scattered examples. If you take a polynomial ring in two or more variables, and mod out by any random prime ideal, you will almost certainly not get a UFD. For example, in the ring k[x,y]/(x^2+y^2-1), x^2 also factors as (1-y)(1+y)
• Abelian groups are a direct sum of cyclic groups. While this is true for finitely-generated abelian groups, it is far from true for infinitely-generated abelian groups, even if you consider infinite direct sums. A typical infinitely-generated abelian group is Q, which cannot be written as a direct sum of any subgroups, but very much not cyclic.
• Finite-dimensional noncommutative division rings over the rationals are all subrings of the quaternions. The definition of the quaternions makes sense with coefficients in any subfield of the reals, and gives you a finite-dimensional division ring over that subfield. If the subfield is itself finite-dimensional over Q, this gives you a finite-dimensional division algebra over Q. I thought that this construction gave you all of the possibilities. This is far from the case. The quaternions are 4-dimensional over their center, but you can construct other division algebras of any dimension over their center, as long as that dimension is a perfect square.

Did this happen to anyone else?

Mathematical Sciences Research Institute (MSRI, usually pronouned “misery”) has put all of the books it has produced since 1995 online. They also have made available Thurston’s 1980 lecture notes on the Geometry and Topology of Three-Manifolds, which has long served as a major source for Thurston’s approach to 3-manifolds.

The American Mathematical Society has put many of its books online. Several of the books available are classics, such as Jacobson’s Structures and Representations of Jordan Algebras, Lefschetz’ Algebraic Topology and Birkhoff’s Dynamical Systems.