数学上，测度(Measure)是一个函数，它对一个给定集合的某些子集指定一个数，这个数可以比作大小、体积、概率等等。

传统的积分是在区间上进行的，后来人们希望把积分推广到任意的集合上，就发展出测度的概念，它在数学分析和概率论有重要的地位。

测度论是实分析的一个分支，研究对象有σ代数、测度、可测函数和积分，其重要性在概率论和统计学中有所体现。

目录[隐藏]• 1 定义• 2 性质o 2.1 单调性o 2.2 可数个可测集的并集的测度o 2.3 可数个可测集的交集的测度• 3 σ有限测度• 4 完备性• 5 例子• 6 自相似分形测度的分维微积分基础引论•7 相关条目•8 参考文献[编辑]定义形式上说，一个测度（详细的说法是可列可加的正测度）是个函数。

设是集合上的一个σ代数，在上定义，于扩充区间中取值，并且满足以下性质：•空集的测度为零：。

•可数可加性，或称σ可加性：若为中可数个两两不交的集合的序列，则所有的并集的测度，等于每个的测度之总和：。

这样的三元组称为一个测度空间，而中的元素称为这个空间中的可测集。

[编辑]性质下面的一些性质可从测度的定义导出：[编辑]单调性测度的单调性：若和为可测集，而且，则。

[编辑]可数个可测集的并集的测度若为可测集（不必是两两不交的），并且对于所有的，⊆，则集合的并集是可测的，且有如下不等式（“次可列可加性”）：以及如下极限：[编辑]可数个可测集的交集的测度若为可测集，并且对于所有的，⊆，则的交集是可测的。

进一步说，如果至少一个的测度有限，则有极限：如若不假设至少一个的测度有限，则上述性质一般不成立（此句的英文原文有不妥之处）。

例如对于每一个，令这里，全部集合都具有无限测度，但它们的交集是空集。

[编辑]σ有限测度详见σ有限测度如果是一个有限实数（而不是），则测度空间称为有限测度空间。

如果可以表示为可数个可测集的并集，而且这些可测集的测度均有限，则该测度空间称为σ有限测度空间。

称测度空间中的一个集合具有σ有限测度，如果可以表示为可数个可测集的并集，而且这些可测集的测度均有限。

作为例子，实数集赋以标准勒贝格测度是σ有限的，但不是有限的。

为说明之，只要考虑闭区间族[k, k+1]，k 取遍所有的整数；这样的区间共有可数多个，每一个的测度为1，而且并起来就是整个实数集。

作为另一个例子，取实数集上的计数测度，即对实数集的每个有限子集，都把元素个数作为它的测度，至于无限子集的测度则令为。

这样的测度空间就不是σ有限的，因为任何有限测度集只含有有限个点，从而，覆盖整个实数轴需要不可数个有限测度集。

σ有限的测度空间有些很好的性质；从这点上说，σ有限性可以类比于拓扑空间的可分性。

[编辑]完备性一个可测集称为零测集，如果。

零测集的子集称为可去集，它未必是可测的，但零测集自然是可去集。

如果所有的可去集都可测，则称该测度为完备测度。

一个测度可以按如下的方式延拓为完备测度：考虑的所有这样的子集，它与某个可测集仅差一个可去集，也就是说与的对称差包含于一个零测集中。

由这些子集生成的σ代数，并定义的值就等于。

[编辑]例子下列是一些测度的例子（重要性与顺序无关）。

•计数测度定义为的“元素个数”。

•一维勒贝格测度是定义在的一个含所有区间的σ代数上的、完备的、平移不变的、满足的唯一测度。

•Circular angle 测度是旋转不变的。

•局部紧拓扑群上的哈尔测度是勒贝格测度的一种推广，而且也有类似的刻划。

•恒零测度定义为，对任意的。

•每一个概率空间都有一个测度，它对全空间取值为1（于是其值全部落到单位区间[0,1]中）。

这就是所谓概率测度。

见概率论公理。

其它例子，包括：狄拉克测度、波莱尔测度、约当测度、遍历测度、欧拉测度、高斯测度、贝尔测度、拉东测度。

[编辑]自相似分形测度的分维微积分基础引论分维微积分在理论基础上主要依据分维导数相对邻近规整导数的位置假设，目前此方法尚不能给出一般函数分维导数的具体解析形式。

分维微积分与分数阶微积分有所不同，分数阶微积分的基础主要依据规整积分变换对分数阶的默认外推，能给出一般函数分数阶微积分的具体形式。

上述这二个研究方向在理论基础上都依赖于规整微积分的表述，但也都缺少严格的证明。

可能的情况是这些表述皆是趋向一个较为基本理论的过渡性近似形式。

而未来可能建立的这个较为基本的理论，将包含更为深刻普适的核心概念定义及基础假设，Newton 微积分将成为其导出结论。

下面的分维微积分主线脉络内容旨在为未来的分维数学解析体系提供前期探讨途径及框架参照。

自相似分形测度的分维微积分计算方法主要是依据上述分维微积分的表述形式，可给出能够直接进行测度计算的方程。

这种方法的分析过程及得到的自相似分形测度与目前普遍采用Hausdorff测度方法（覆盖方法）得到的结果不同，覆盖方法分析过程较为复杂，得到的测度一般依赖于所使用的覆盖方式及迭代技巧，计算方法的普适性较弱。

[1] [编辑]相关条目•外测度(Outer measure)•几乎处处(Almost everywhere)•勒贝格测度(Lebesgue measure)[编辑]参考文献1.^/papers/paper-pdf/celestialand maths-pdf.pdf/spires/find/hep/www?j=005 45,22,451/abs/2007PrGeo..22..451Y•R. M. Dudley, 2002. Real Analysis and Probability.Cambridge University Press.• D. H. Fremlin, 2000. Measure Theory. Torres Fremlin. •Paul Halmos, 1950. Measure theory. Van Nostrand and Co.•M. E. Munroe, 1953. Introduction to Measure and Integration. Addison Wesley.•Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8.Emphasizes the Daniell integral.•阎坤. 天体运行轨道的背景介质理论导引与自相似分形测度计算的分维微积分基础[J]. 地球物理学进展，2007，22(2)：451～462. YAN Kun. Introduction on background medium theory about celestial body motion orbit and foundation of fractional-dimension calculus about self-similar fractalmeasure calculation[J]. Progress in Geophysics(inChinese with abstract in English)，2007，22(2)：451～462. 取自"/wiki/%E6%B5%8B%E5%BA%A6" 2个分类: 测度论| 数学结构MeasureIn mathematics, more specifically measure theory, a measure is intuitively a certain association between subsets of a given set X and the (extended set) of non-negative real numbers. Often, some subsets of a given set X are not required to be associated to a non-negative real number; the subsets which are required to be associated to a non-negative real number are known as the measurable subsets of X. The collection of all measurable subsets of X is required to form what is known as a sigma algebra; namely, a sigma algebra is a subcollection of the collection of all subsets of X that in addition, satisfies certain axioms.Measures can be thought of as a generalization of the notions: 'length,' 'area' and 'volume.' The Lebesgue measure defines this for subsets of a Euclidean space, and an arbitrary measuregeneralizes this notion to subsets of any set. The original intent for measure was to define the Lebesgue integral, which increases the set of integrable functions considerably. It has since found numerous applications in probability theory, in addition to several other areas of academia, particularly in mathematical analysis. There is a related notion of volume form used in differential topology.Contents[hide]• 1 Definition• 2 Propertieso 2.1 Monotonicityo 2.2 Measures of infinite unions of measurable setso 2.3 Measures of infinite intersections of measurable sets• 3 Sigma-finite measures• 4 Completeness• 5 Examples• 6 Non-measurable sets•7 Generalizations•8 See also•9 References•10 External links[edit] DefinitionFormally, a measure μ is a function (usually denoted by a Greek letter such as μ) defined on a σ-algebra Σ over a set X and taking values in the extended interval [0,∞] such that the following properties are satisfied:•The empty set has measure zero:•Countable additivity or σ-additivity: if E1, E2, E3, … is a countable sequence of pairwise disjoint sets in Σ, themeasure of the union of all the E i is equal to the sum of the measures of each E i:The triple (X, Σ, μ) is then called a measure space, and the members of Σ are called measurable sets.A probability measure is a measure with total measure one (i.e., μ(X) = 1); a probability space is a measure space with a probability measure.For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures. Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous functions with compact support. This approach is taken by Bourbaki (2004) and a number of other authors. For more details see Radon measure.[edit] PropertiesSeveral further properties can be derived from the definition of a countably additive measure.[edit] MonotonicityA measure μ is monotonic: If E1 and E2 are measurable sets with E1⊆ E2 then[edit] Measures of infinite unions of measurable setsA measure μ is countably subadditive: If E1, E2, E3, … is a countable sequence of sets in Σ, not ne cessarily disjoint, thenA measure μ is continuous from below: If E1, E2, E3, … are measurable sets and E n is a subset of E n + 1 for all n, then the union of the sets E n is measurable, and[edit] Measures of infinite intersections of measurable setsA measure μ is cont inuous from above: If E1, E2, E3, … are measurable sets and E n + 1 is a subset of E n for all n, then the intersection of the sets E n is measurable; furthermore, if at least one of the E n has finite measure, thenThis property is false without the assumption that at least one of the E n has finite measure. For instance, for each n∈ N, letwhich all have infinite measure, but the intersection is empty. [edit] Sigma-finite measuresMain article: Sigma-finite measureA measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞). It is called σ-finite if X can be decomposed into a countable union of measurable sets of finite measure. A set in a measure space has σ-finite measure if it is a countable union of sets with finite measure.For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals [k,k+1] for all integers k; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the Lindelöf property of topological spaces. They canbe also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.[edit] CompletenessA measurable set X is called a null set if μ(X)=0. A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set is measurable.A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set. One defines μ(Y) to equal μ(X). [edit] ExamplesSome important measures are listed here.•The counting measure is defined by μ(S) = number of elements in S.•The Lebesgue measure on R is a completetranslation-invariant measure on a σ-algebra containingthe intervals in R such that μ([0,1]) = 1; and every othermeasure with these properties extends Lebesgue measure.•Circular angle measure is invariant under rotation.•The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure (and also ofcounting measure and circular angle measure) and hassimilar uniqueness properties.•The Hausdorff measure which is a refinement of the Lebesgue measure to some fractal sets.•Every probability space gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the unit interval [0,1]). Such a measure is calleda probability measure. See probability axioms.•The D irac measure μa (cf. Dirac delta function) is given by μa(S) = χS(a) = [a∈ S], where χS is the characteristicfunction of S and the brackets signify the Iverson bracket.The measure of a set is 1 if it contains the point a and 0otherwise.Other 'named' measures include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gauss measure, Baire measure, Radon measure.[edit] Non-measurable setsMain article: Non-measurable setIf the axiom of choice is assumed to be true, not all subsets of Euclidean space are Lebesgue measurable; examples of such sets include the Vitali set, and the non-measurable sets postulated by the Hausdorff paradox and the Banach–Tarski paradox.[edit] GeneralizationsFor certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure. Measures that take values in Banach spaces have been studied extensively. A measure that takes values in the set of self-adjoint projections on a Hilbert space is called a projection-valued measure; these are used mainly in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term "positive measure" is used.Another generalization is the finitely additive measure. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first, but proved to be not so useful. It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of L∞ and theStone–Čech co mpactification. All these are linked in one way or another to the axiom of choice.The remarkable result in integral geometry known as Hadwiger's theorem states that the space of translation-invariant, finitely additive, not-necessarily-nonnegative set functions defined on finite unions of compact convex sets in R n consists (up to scalar multiples) of one "measure" that is "homogeneous of degree k" for each k = 0, 1, 2, ..., n, and linear combinations of those "measures". "Homogeneous of degree k" means that rescaling any set by any factor c > 0 multiplies the set's "measure" by c k. The one that is homogeneous of degree n is the ordinaryn-dimensional volume. The one that is homogeneous of degree n−1 is the "surface volume". The one that is homogeneous of degree 1 is a mysterious function called the "mean width", a misnomer. The one that is homogeneous of degree 0 is the Euler characteristic.A measure is a special kind of content.[edit] See alsoLook up measurable inWiktionary, the free dictionary.•Outer measure•Inner measure•Hausdorff measure•Product measure•Pushforward measure•Lebesgue measure•Vector measure•Almost everywhere•Lebesgue integration•Caratheodory extension theorem•Measurable function•Geometric measure theory•Volume form[edit] References•R. G. Bartle, 1995. The Elements of Integration and Lebesgue Measure. Wiley Interscience.•Bourbaki, Nicolas (2004), Integration I, Springer Verlag, ISBN 3-540-41129-1 Chapter III.•R. M. Dudley, 2002. Real Analysis and Probability.Cambridge University Press.•Folland, Gerald B. (1999), Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons,ISBN 0-471-317160-0 Second edition.• D. H. Fremlin, 2000. Measure Theory. Torres Fremlin.•Paul Halmos, 1950. Measure theory. Van Nostrand and Co.•R. Duncan Luce and Louis Narens (1987). "measurement, theory of," The New Palgrave: A Dictionary of Economics, v.3, pp. 428-32.•M. E. Munroe, 1953. Introduction to Measure and Integration. Addison Wesley.•Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8.Emphasizes the Daniell integral.[edit] External links•Measure theory for dummies, pdf articleRetrieved from"/wiki/Measure_(mathematics)" Categories: Mathematical structures | Measure theory | Measures (measure theory)。