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
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
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
- 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
- Is a sigma-algebra of sets also an algebra?
- When is the Borel sigma algebra on E x E the square of that on E?
- Regular signed Borel measures
- Some examples of non-Borel (measurable) sets.
- Constructing a non-measurable subset of R^1
- Can we recognize the Borel sets among the measurable ones?
- Finitely-additive measures on R^n (which are not countably additive); the Banach-Tarski paradox.
- Implications of Axiom of (Dependent) Choice, Axiom of Determinateness, and the existence of inaccessible cardinals, for analysis (existence of non-measurable sets)
- Assured existence of uncountable measure zero subsets is undecidable
- Product measures on the unit square
- What, exactly, does continuity almost everywhere mean?
- Convergence of signed measures does not imply convergence of positive parts
- 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.
- Standard definition of Riemann integral
- Different types of integrals (Riemann, Lebesgue, etc.)
- Pointers, summaries of articles on history of types of integration
- 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.
- The Borel-Cantelli lemma: iterates of measure-preserving maps escape sets of finite measure
- What is the area of the Mandelbrot set?
- Invariant measures (cylindrical, Wiener) on infinite-dimensional sphere
- Defining "measure zero" on infinite dimensional spaces (prevalence)
- Hamel bases of R over Q cannot be 'nice' (Borel, etc.)
- 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.
- Proving some functions have no elementary antiderivative [update of previous article]
- 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.
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