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# 28: Measure and integration

## Introduction

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.

## Applications and related fields

For numerical integration of real functions see Numerical Analysis

Treated here are measure theory both abstractly and on the real line. For measure theory and analysis on Lie groups, see 43-XX. For measure and integration on infinite-dimensional vector spaces see 46-XX and 47-XX.

The Borel sets and related families are constructed as a part of "descriptive" set theory (now in section 03E).

Chaotic attractors are treated in 37: Dynamical Systems; these may lead to fractal sets.

Many common and important indefinite integrals cannot be expressed in terms of the elementary functions but are themselves studied in 33: Special Functions. This includes the elliptic functions, the gamma function, the Fresnel integrals, and so on. (Indeed, many of these functions are defined by integrals).

There is a general theory of computing anti-derivatives in "closed form"; this isn't really part of the study of integration at all. See rather 12H: Differential and difference algebra

## Subfields

• 28A: Classical measure theory
• 28B: Set functions, measures and integrals with values in abstract spaces
• 28D: Measure-theoretic ergodic theory, see also 11K50, 11K55, 22D40, 47A35, 54H20, 58FXX, 60FXX, 60G10
• 28E: Miscellaneous topics in measure theory

Browse all (old) classifications for this area at the AMS.

## Textbooks, reference works, and tutorials

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.

## Software and tables

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