測度的概念和相關

1、數(shù)學上，測度(Measure)是一個函數(shù)，它對一個給定集合的某些子集 指定一個數(shù)，這個數(shù)可以比作大小、體積、概率等等。傳統(tǒng)的積分是 在區(qū)間上進行的，后來人們希望把積分推廣到任意的集合上，就發(fā)展 出測度的概念，它在數(shù)學分析和概率論有重要的地位。測度論是實分析的一個分支，研究對象有0代數(shù)、測度、可測函數(shù)和 積分，其重要性在 概率論和統(tǒng)計學中有所體現(xiàn)。目錄隱藏? 1定義? 2性質(zhì)0 2.1單調(diào)性0 2.2可數(shù)個可測集的并集的測度o 2.3可數(shù)個可測集的交集的測度? 3 0有限測度? 4完備性? 5例子? 6自相似分形測度的分維微積分基礎引論? 7相關條目? 8參考文獻編輯定義形式上說，一個測度網(wǎng)（詳

2、細的說法是可列可加的正測度）是個函數(shù)。 設A是集合X上的一個"代數(shù)，"在上/定義，于擴充區(qū)間網(wǎng)8】中取 值，并且滿足以下性質(zhì)：?空集的測度為零：岫=0。?可數(shù)可加性，或稱（T可加性：若%，石2, ,為4中可數(shù)個兩兩 不交的集合的序列，則所有后的并集的測度，等于每個居I的測 度之總和：88】=自。這樣的三元組（X，4 M）稱為一個測度空間，WA中的元素稱為這個空 間中的可測集。編輯性質(zhì)下面的一些性 質(zhì)可從測度的定義導出：編輯單調(diào)性測度的單調(diào)性：若已和石2為可測集，而且鼻口 E”則gW不。編輯可數(shù)個可測集的并集的測度若& 一為可測集（不必是兩兩不交的），并且對于所有的n

3、., 品?則集合&的并集是可測的，且有如下不等式（“次可列可加性”）：ku居”愛國 £=1W=1以及如下極限：（U 居）=Um一二RTCK=1編輯可數(shù)個可測集的交集的測度若E1,已2,一,為可測集，并且對于所有的巴則昂的交集是可測的。進一步說，如果至少一個石R的測度有限，則有極限:（&） = Um!tcc1=1如若不假設至少一個島的測度有限，則上述性質(zhì)一般不成立（此句的英文原文有不妥之處）。例如對于每一個ftEN,令En =卜更 oo） C R這里，全部集合都具有無限 測度，但它們的交集是空集。編輯。有限測度詳見"有限測度如果可是一個有限實數(shù)（而不是8）,則

4、測度空間（X,4浦稱為有 限測度空間。如果a可以表示為可數(shù)個可測集的并集，而且這些可測 集的測度均有限，則該測度空間稱為。有限測度空間。稱測度空間中 的一個集合3具有。有限測度，如果及可以表示為可數(shù)個可測集的并 集，而且這些可測集的測度均有限。作為例子，實數(shù)集賦以標準勒貝格測度是。有限的，但不是有限的。為說明之，只要考慮閉區(qū)間族k, k+1 , k取遍所有的整數(shù)；這樣的 區(qū)間共有可數(shù)多個，每一個的測度為1,而且并起來就是整個實數(shù)集。 作為另一個例子，取實數(shù)集上的計數(shù)測度，即對實數(shù)集的每個有限子 集，都把元素個數(shù)作為它的測度，至于無限子集的測度則令為這 樣的測度空間就不是。有限的，因為任何有限測

5、度集只含有有限個 點，從而，覆蓋整個實數(shù)軸需要不可數(shù)個有限測度集。）有限的測度 空間有些很好的性質(zhì)；從這點上說，）有限性可以類比于拓撲空間的 可分性。編輯完備性一個可測集2V稱為零測集，如果以N）=0。零測集的子集稱為可去 集，它未必是可測的，但零測集自然是可去集。如果所有的可去集都 可測，則稱該測度為完備測度。一個測度可以按如下的方式延拓 為完備測度：考慮X的所有這樣的子 集F,它與某個可測集F僅差一個可去集，也就是說E與*的對稱差 包含于一個零測集中。由這些子集F生成的0代數(shù)，并定如（口）的值 就等于編輯例子下列是一些測度的例子（重要性 與順序無關）。?計數(shù)測度定義為似M = S的“元素個

6、數(shù)”。? 一維勒貝格測度 是定義在R的一個含所有區(qū)間的）代數(shù)上的、 完備的、平移不變的、滿網(wǎng)（口，1）= 1的唯一測度。? Circular angle 測度 是旋轉(zhuǎn)不變的。?局部緊拓撲群上的哈爾測度是勒貝格測度的一種推廣，而且也 有類似的刻劃。?恒零測度定義為以S）= °,對任意的S。?每一個概率空間都有一個測度，它對全空間取值為1 （于是其 值全部落到單位區(qū)間0,1中）。這就是所謂概率測度。見概率 論公理。其它例子，包括：狄拉克 測度、波萊爾測度、約當測度、遍歷測度、 歐拉測度、高斯測度、貝爾測度、拉東測度。編輯自相似分形測度的分維微積分基礎引論分維微積分在理論基礎上主要依據(jù)分維

7、導數(shù)相對鄰近規(guī)整導數(shù)的位 置假設，目前此方法尚不能給出一般函數(shù)分維導數(shù)的具體解析形式。 分維微積分與分數(shù)階微積分有所不同，分數(shù)階微積分的基礎主要依據(jù) 規(guī)整積分變換對分數(shù)階的默認外推，能給出一般函數(shù)分數(shù)階微積分的 具體形式。 上述這二個研究方向在理論基礎上都依賴于規(guī)整微積分 的表述，但也都缺少 嚴格的證明。可能的情況是這些表述皆是趨向 一個較為基本理論的過渡性近似形式。而未來可能建立的這個較為基 本的理論，將包含更為深刻普適的核心概念定義及基礎假設，Newton 微積分將成為其導出結(jié)論。下面的分維微積分主線脈絡內(nèi)容旨在為 未來的分維數(shù)學解析體系提供前期探 討途徑及框架參照。自相似分 形測度的分維

8、微積分計算方法主要是依據(jù)上述分 維微積分的表述形 式，可給出能夠直接進行測度計算的方程。這種方法的分析過程及 得到的自相似分形測度與目前普遍采用Hausdorff測度方法(覆蓋方 法)得到的結(jié)果不同，覆蓋方法分析過程較為復雜，得到的測度一般 依賴于所使用的覆蓋方式及迭代技巧，計算方法的普適性較弱。1編輯相關條目? 外測度(Outer measure)? 幾乎處處(Almost everywhere)?勒貝格測度(Lebesgue measure)編輯參考文獻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.?閻坤.天體運行軌道的背景介質(zhì)理論導引與自相似分形測度計 算的分維微積分基礎J.地球物理學進展，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個分類：測度論|數(shù)學結(jié)構(gòu)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兇.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