This reference list will grow over time. I intend it to contain all the stuff I look at from time to time as well as comments that might help someone also interested in learning from the references. See also the Notes and Comments subpage for comments on, and links to, various papers and notes of geometric measure theory and geometric analysis and their applications.

Measure Theory and Fine Properties of Functions, L. Craig Evans and Ronald F. Gariepy. This text is the best reference to anything it contains. In particular it does a beautiful job covering the theory of BV functions and Cacioppoli sets. This theory was introduced by De Giorgi in 1960 to solve co-dimension 1 minimal surface problems. Other topics covered including measures, densities, covering theorems, the area and co-area formulas, the theory of Sobolev functions, the Whitney extension theorem, and other useful and usually-not-covered topics.

Geometry of Sets and Measures in Euclidean Spaces, Pertti Mattila. This is the reference for the rectifiability, fractal measure, etc branch of the subject. Nothing about currents and varifolds here. But it is a favorite with students for its excellent exposition.

GMT, Herbert Federer. This is the standard, comprehensive (but in some places dated) reference for the subject. It is notoriously difficult in spots. Precision and generality (and often you need it) are characteristics of the text. Remembering that what is being written about is most often geometric can help reading it. Frank Morgan's text was constructed as an interface to this text.

Lectures on GMT, Leon Simon. A more readable GMT text that does get to the details, but is not of course comprehensive. This is the text that many people first learn GMT from (often in combination with an initial read of Frank Morgan's text).

Seminar on GMT, Bob Hardt and Leon Simon. Nice, fairly brief notes from a 10 lecture short course taught by Bob and Leon in Dusseldorf in 1984. This is a nice first look at th esubject.

GMT, Fanghua Lin and Xiaoping Yang. This is a rich reference that is more up to date in some ways than Federer's book. It is not as complete of course and it has frequent typos and even some typos that seem like errors. On the other hand, Fanghua told me that he had sent the publisher corrections which they refused to use to update the book. With a bit more background, it is a great reference.

Riemannian Geometry, Manfredo Do Carmo. My favorite book on Riemannian Geomtry. Beautifully written. Really gives you a feel for the tools and the intuitions without holding your hand too much. And it is thin!