A measure is a nonnegative, countably additive function defined on a collection of sets, usually a sigma-algebra. The precise definitions of measure and measurability provide a rigorous basis for integration. These definitions also underlie much of analysis and calculus.

For example, every definition of an integral is based on a particular measure: the Riemann integral is based on Jordan measure, and the Lebesgue integral is based on Lebesgue measure. The study of measures and their application to integration is known as measure theory.

More generally, a measure on a delta-ring takes values in and satisfies

where is the empty set, and

for any finite or countable collection of pairwise disjoint sets in such that is also in .

If the underlying space is a countable union of sets in of finite measure, then can be extended uniquely to a measure defined on the sigma-algebra generated by .

is said to be a probability measure on a set if and is a sigma-algebra.

In a special set-theoretic convention, a nontrivial countably additive measure is a probability measure on the full power set of an infinite set that vanishes on every singleton set (Jech 1997). This convention imposes additional requirements beyond those of an ordinary probability measure, which need only be defined on a sigma-algebra and may assign positive measure to singleton sets. The special convention requires:

1. and ,

2. If then ,

3. for all (nontriviality),

4. If , , 1, 2, ..., are pairwise disjoint, then

Such a measure cannot exist on a countably infinite set, since countable additivity would give .

A measure may be extended by completion. All subsets of sets in with measure zero form a delta-ring . By "changing" sets in on a set from , a delta-ring which is the completion of with respect to is obtained.

The measure is called a complete measure if . If is not complete, it may be extended to by setting , where and .

See also

Almost Everywhere, Borel Measure, Complete Measure, Counting Measure, Ergodic Measure, Euler Measure, Gauss Measure, Haar Measure, Hausdorff Measure, Helson-Szegö Measure, Integral, Jordan Measure, Lebesgue Measure, Liouville Measure, Mahler Measure, Measurable Space, Measure Algebra, Measure Space, Minkowski Measure, Natural Measure, Probability Measure, Radon Measure, Signed Measure, Wiener Measure Explore this topic in the MathWorld classroom

Portions of this entry contributed by Allan Cortzen

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References

Czyz, J. Paradoxes of Measures and Dimensions Originating in Felix Hausdorff's Ideas. Singapore: World Scientific, 1994.Jech, T. J. Set Theory, 2nd ed. Berlin, Germany: Springer-Verlag, p. 295, 1997.

Referenced on Wolfram|Alpha

Measure

Cite this as:

Weisstein, Eric W., with contributions by Allan Cortzen. "Measure." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Measure.html

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