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A measure is a unit of time in Western music, also known as the bar. It represents a regular grouping of beats?, as indicated in notation by the time signature. |
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A measure is a unit of time in Western music, also known as a bar. It represents a regular grouping of beat?s, as indicated in notation by the time signature. |
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In mathematics: |
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In mathematics: |
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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. |
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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. |
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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)=&infinity; otherwise. |
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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. |
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Some important measures are listed here. The Lebesgue measure is the unique 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. |
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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. |
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See Probability axioms |
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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. |
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