28: Measure and integration
Measure theory and integration is the study of lengths, surface area, and volumes in general spaces. This is a critical feature of a full development of integration theory; moreover, it provides the basic framework for probability theory. Measure theory is a meeting place between the tame applicability of real functions and the wild possibilities of set theory. This is the setting of fractals.
For numerical integration of real functions see Numerical Analysis
For analysis on manifolds, See 58-XX; in particular, information on chaos and fractal sets (especially obtained through dynamical systems) is there. For analysis on Lie groups, see 43-XX.
The Borel sets and related families are constructed as a part of "descriptive" set theory
Integrals of certain fairly explicit functions (e.g. elliptic integrals) may be expressed in terms of various special functions. (Indeed, many of the latter are defined by integrals).
Browse all (old) classifications for this area at the AMS.
Bourbaki, N., "Integration", separate chapters published separately by Hermann, Paris ca. 1969
Bear, H. S.: "A primer of Lebesgue integration", Academic Press, Inc., San Diego, CA, 1995. 163 pp. ISBN 0-12-083970-9 MR96f:28001
Cohn, Donald L.: "Measure Theory", Birkhäuser Boston, Inc., Boston, MA, 1993. 373 pp. ISBN 0-8176-3003-1 MR98b:28001 (Reprint of the 1980 original: see MR81k:28001.)
Ulam, S. M.: "What is measure?", Amer. Math. Monthly 50, (1943). 597--602. MR5,113g
Birkhoff, G. D.: "What is the ergodic theorem?" Amer. Math. Monthly 49, (1942). 222--226. MR4,15b
Schanuel, Stephen H.: "What is the length of a potato? An introduction to geometric measure theory" Categories in continuum physics (Buffalo, N.Y., 1982), 118--126; Lecture Notes in Math., 1174, Springer, Berlin, 1986.
There is a newsgroup sci.fractals; there is a (somewhat dated!) Fractal FAQ for it.
Handbooks of integrals are common; particularly massive is the set of integral tables by Gradshteyn, I.S. and Ryzhik, I.M. "Tables of Integrals, Series, and Products", (5th ed, 1993), San Diego CA: Academic Press. Somewhat closer to a textbook (offering some discussion of the principal themes) is Zwillinger, Daniel: "Handbook of integration", Jones and Bartlett Publishers, Boston, MA, 1992. ISBN 0-86720-293-9.
Online integrators from Wolfram Inc. and Fateman. (The latter calls the former if it gets stuck.)
The GAMS software tree has a node for numerical evaluation of definite integrals