测度的概念和相关

1、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0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0 2.1单调性0 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 0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，一个测度网（详

2、细的说法是可列可加的正测度）是个函数。 设A是集合X上的一个"代数，"在上/定义，于扩充区间网8】中取 值，并且满足以下性质：?空集的测度为零：岫=0。?可数可加性，或称（T可加性：若%，石2, ,为4中可数个两两 不交的集合的序列，则所有后的并集的测度，等于每个居I的测 度之总和：88】=自。这样的三元组（X，4 M）称为一个测度空间，WA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：若已和石2为可测集，而且鼻口 E”则gW不。编辑可数个可测集的并集的测度若& 一为可测集（不必是两两不交的），并且对于所有的n

3、., 品?则集合&的并集是可测的，且有如下不等式（“次可列可加性”）：ku居”爱国 £=1W=1以及如下极限：（U 居）=Um一二RTCK=1编辑可数个可测集的交集的测度若E1,已2,一,为可测集，并且对于所有的巴则昂的交集是可测的。进一步说，如果至少一个石R的测度有限，则有极限:（&） = Um!tcc1=1如若不假设至少一个岛的测度有限，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ftEN,令En =卜更 oo） C R这里，全部集合都具有无限 测度，但它们的交集是空集。编辑。有限测度详见"有限测度如果可是一个有限实数（而不是8）,则

4、测度空间（X,4浦称为有 限测度空间。如果a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。有限测度空间。称测度空间中 的一个集合3具有。有限测度，如果及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。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。有限的，因为任何有限测

5、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。）有限的测度 空间有些很好的性质；从这点上说，）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2V称为零测集，如果以N）=0。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X的所有这样的子 集F,它与某个可测集F仅差一个可去集，也就是说E与*的对称差 包含于一个零测集中。由这些子集F生成的0代数，并定如（口）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）。?计数测度定义为似M = S的“元素个

6、数”。? 一维勒贝格测度 是定义在R的一个含所有区间的）代数上的、 完备的、平移不变的、满网（口，1）= 1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。?恒零测度定义为以S）= °,对任意的S。?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

7、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

8、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1.八 maths-pdf.pdf/spires/find/hep/www?j=00

9、5 45,22,451http:/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

10、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. Y

11、AN Kun. Introduction on background medium theory about celestial body motion orbit and foundation of fractional-dimension calculus about self-similar fractalmeasure calculationJ. Progress in Geophysics(inChinese with abstract in English) , 2007 , 22(2) : 451 462.取自"/wiki/%

12、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

13、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 ofX. The collection of all measurablesubsets of X is required to form what is known as a sigma algebra; namely, a sigma algebra is a subcollection of the coll

14、ection of all subsets ofX that in addition, satisfies certainaxioms.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 measure generalizes this noti

15、on 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 an

16、alysis. There is a related notion of volume form used in differential topology.Contentshide? 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 Example

17、s? 6 Non-measurable sets? 7 Generalizations? 8 See also? 9 References? 10 External linksedit DefinitionFormally, a measure is afunction (usually denoted by a Greek letter such as ) defined on a -algebra E over a set X and taking values in the extended interval 0, 0° such that the following prop

18、erties are satisfied:? The empty set has measure zero:? Countable additivity or o-additivity: if Ei, E2, E3, is acountable sequence of pairwise disjoint sets in E , the measure of the union of all the Ei is equal to the sum of themeasures of eachEi：The triple ( X, E ,) is then called a measure space

19、 , and the members of E are called measurable setsA probability measure is a measure with total measure one (i.e., wK) = 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 me

20、asure 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

21、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 is monotonic: If Ei and E2 are measurable sets with Ei ? E2 thenedit Meas

22、ures of infinite unions of measurable setsA measure is countably subadditive: If Ei, E2, E3, is a countable sequence of sets in E , not necessarily disjoint, then(g 30U eJ < £可用.A measure is continuous from below: If Ei, E2, E3, are measurable sets and En is a subset of En + 1 for all n, the

23、n the union of the sets En is measurable, and=Jim皿居）第一8edit Measures of infinite intersections of measurable setsA measure is cont inuous from above: If Ei, E2, E3, are measurable sets and En + 1 is a subset of En for all n, then the intersection of the sets En is measurable; furthermore, if at leas

24、t one of the En has finite measure, thenM (0居)=吧ME)This property is false without the assumption that at least one of the En has finite measure. For instance, for each n N, letEn = n. 00) C R which all have infinite measure, but the intersection is empty.edit Sigma-finite measuresMain article: Sigma

25、-finite measureA measure space ( X, E ,) is called finite if X) is a finite realnumber (rather than 0°). It is called o-finite if X can be decomposed into a countable union of measurable sets of finite measure. A set in a measure space haso-finite measure if it is acountable union of sets with

26、finite measure.For example, the real numbers with the standard Lebesgue measure are (-finite but not finite. Consider the closed intervals k,k+1 for all integersk; there are countably many suchintervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real nu

27、mbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not -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 -fi

28、nite measure spaces have some very convenient properties; -finiteness can be compared in this respect to the Lindel?f property of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.edit CompletenessA measu

29、rable set X is called a null set if 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 co

30、mplete one by considering the -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 definesY) tqtequal凶.edit ExamplesSome important measures are listed here.? The counting measure is defi

31、ned byS) = number ofelements in S.? The Lebesgue measure on R is a complete translation-invariant measure on ao-algebra containing the intervals in R such that (0,1) = 1; and every other measure with these properties extends Lebesgue measure.? Circular angle measure is invariant under rotation.? The

32、 Haar measure for a locally compact topological group is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties.? The Hausdorff measure which is a refinement of the Lebesgue measure to some fractal sets.? Every probabi

33、lity 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 called a probability measure . See probability axioms.? The D irac measure a (cf. Dirac delta function) is given by 出(S) = S(a) = a S, where

34、力 is the characteristic function of S and the brackets signify the Iverson bracket. The measure of a set is 1 if it contains the point a and 0 otherwise.Other 'named' measures include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gauss measure, Baire measure, Radon measure.

35、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-Tarskiparadox

36、.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 wit

37、h 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 an

38、alysis 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 re

39、quiring 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 the Stone ?ech co mpactification.

40、 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 i

41、n Rn 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 degreek" means that rescalingany set by any factor c > 0 multiplies the set'

42、s "measure" byc k.The one that is homogeneous of degree n is the ordinary n-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.K'iLriiirsyrii rifir hi M 彳 m AiB-Mfpedit See alsoLook up measurable inWiktionary, the free dict