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[Texts]## 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).

- 28A: Classical measure theory
- 28B: Set functions, measures and integrals with values in abstract spaces
- 28C: Set functions and measures on spaces with additional structure, see also 46G12, 58C35, 58D20
- 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.

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

- Here are the AMS and Goettingen resource pages for area 28.

- Some examples of non-Borel (measurable) sets.
- Finitely-additive measures on R^n (which are not countably additive); the Banach-Tarski paradox.
- Connection between fractals and Newton's method.
- The set of all fractions m!/2^n is a dense subset of the real line.
- Is the Mandelbrot set measurable?
- Computer code to draw the Mandelbrot set.
- Can we recognize the Borel sets among the measurable ones?
- What, exactly, does continuity almost everywhere mean?
- Conditions necessary for an application of Fubini's theorem (interchange order of integration).
- Failure of Fubini's theorem when the integrand is not integral over the rectangle.
- Proof of Fatou's Lemma (convergence a.e. of a sequence of functions implies convergence of the integrals).
- Can one reconstruct a function knowing all integrals over balls of radius 1 ?
- Do the integrals of a function over rectangles determine that function uniquely?
- Do the integrals of a function over triangles determine F?
- The method of stationary phase for computing integrals of oscillatory functions.
- Applications of Stieltjes integrals.
- Is there a closed-form "solution" for an elliptic integral? (no)
- What functions have antiderivatives which are elementary functions ? Citations and long article by Matthew Wiener. (Includes topics in symbolic integration.) Frequently-mentioned integrands with no elementary antiderivative include exp(-x^2), sin(x)/x, x^x, sqrt(1-x^4), and many variants.
- Comparisons of deterministic and heuristic algorithms to decide whether a function has an elementary antiderivative (Risch algorithm, popular software, etc.)
- Definite integrals which lead to values of the zeta function.

Last modified 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org