Measure

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To perform a measurement.

Below are some specialised meanings of measure.


In music:

A measure is a unit of time in Western music, also known as a bar. It represents a regular grouping of beats, as indicated in notation by the time signature.


In mathematics:

A measure is a countably additive (see below) set function m over a sigma algebra, which takes non-negative (but possibly infinite) values. Sets in the sigma algebra are called "m-measurable" or "measurable" for short. To avoid degeneracy, we request that m(0)=0 (the measure of the empty set is zero). For certain purposes, it is useful to have a "measure" whose image is not a non-negative real nor infinity, in which case countable additivity only is preserved. For instance, a countably additive set function with values in the (signed) real numbers is called a charge, while a measure with values in the complex numbers is called a complex measure. A measure that takes values in a Banach space is called a spectral measure, and is used mainly in functional analysis for the spectral theorem. Finally, a measure which takes values in the unit interval [0,1] is called a probability measure.

The countable additivity requirement can be written as follows:

(Countable Additivity): if E1, E2, E3, ... are pairwise disjoint measurable sets and E is their union then the measure m(E) is equal to the sum ∑m(Ek).

If m is a non-negative measure then the sum is unambiguous. Otherwise there is the possiblity that the sum would fail to converge, and more work would be necessary to calm our worries.

It is important to note that finite additivity is insufficient. A counter example over the integers (the sigma algebra is the power set) is the "measure" m which has value m(S)=0 whenever S is a finite set and m(S)=∞ otherwise.

A measurable set S is called a null-set if m(S) = 0. The measure m is called complete if every subset of a null-set is measurable and itself a null-set.

Some important measures are listed here. The Lebesgue measure is the unique complete translation invariant measure on a sigma algebra containing the intervals in R such that m([0,1])=1. The counting measure is define by m(S)=number of elements in S. The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure and has a similar uniqueness property. The zero measure is defined by m(S)=0 for all S.

See also probability axioms.