查看“︁幾乎所有”︁的源代码

數學中，'''幾乎所有'''（{{lang-en|'''Almost all'''}}）表示「除了極少數可忽略的以外，其他都是」。更準確的說法，若<math>X</math>是[[集合 (数学)|集合]]，「集合<math>X</math>中幾乎所有的元素」表示「集合<math>X</math>中，不考慮在某個{{link-en|可忽略集合|negligible set|可忽略}}[[子集]]內元素的其他元素。」「可忽略」的具體意思則依上下文而定，可能是[[有限集合]]、[[可數集]]或[[零测集]]。

相反的，'''幾乎沒有'''（almost no）表示「只有極少數可忽略的是」，「集合<math>X</math>中幾乎沒有的元素」表示「集合<math>X</math>中，只有在某個可忽略子集內的元素」。

==不同數學領域中的意思==
===普遍的意思===
<!--{{further|Cofinite set}}-->
數學裡的「幾乎所有」有時會指「[[无限集合]]中的元素，只有[[有限集合|有限多]]個不符合，其餘都符合」的情形{{r|Cahen1|Cahen2}}。此用法也會用在哲學上{{r|Gardenfors}}。「幾乎所有」也可以指「[[不可數集]]中的元素，只有[[可數集|可數數量]]的不符合，其餘都符合」的情形{{r|Schwartzman|group=sec}}。

例如：
* 幾乎所有正實數都超過10<sup>12</sup>{{r|Courant|page=293}}。
* 幾乎所有[[质数]]都是奇数（只有2例外）<ref>{{Cite book|last1=Movshovitz-hadar|first1=Nitsa|url=https://books.google.com/books?id=lp15DwAAQBAJ&q=Almost+all+prime+numbers+are+odd&pg=PA38|title=Logic In Wonderland: An Introduction To Logic Through Reading Alice's Adventures In Wonderland - Teacher's Guidebook|last2=Shriki|first2=Atara|date=2018-10-08|publisher=World Scientific|isbn=978-981-320-864-3|pages=38|language=en|quote=This can also be expressed in the statement: 'Almost all prime numbers are odd.'}}</ref>
* 幾乎所有[[多面体]]都是非[[正多面體]]（只有九個例外，五個[[柏拉圖立體]]和四個[[星形正多面體]]）。
* 若<var>P</var>是非零[[多項式]]，則<var>P(x)</var> &ne; 0對幾乎所有的<var>x</var>都成立。
===量測理論中的意思===
{{further|幾乎處處}}
[[File:CantorEscalier.svg|thumb|right|250px|[[康托尔函数]]是幾乎處處導數都為零的函數]]

在探討[[实数]]時，有時「幾乎所有」是指「除了在某個[[零测集]]以外的所有實數。」{{r|Korevaar|Natanson}}{{r|Clapham|group=sec}}。同様地，若<var>S</var>是某個實數集合，則「幾乎所有在集合<var>S</var>裡的數字」是指「除了在某個[[零测集]]以外，集合<var>S</var>的所有實數。」{{r|Sohrab}}[[數線]]可以視為是一維的[[欧几里得空间]]。在更廣義的<var>n</var>維空間（<var>n</var>為正整數），其定義則推廣為「除了在某個零测集以外，空間裡的所有點。」{{r|James|group=sec}}或是「除了在某個零测集以外，集合<var>S</var>裡的所有點。」 （此時，<var>S</var>是空間中點的集合）{{r|Helmberg}}。更廣義的說法，「幾乎所有」在[[测度]]理論中有時是指[[幾乎處處]]{{r|Vestrup|Billingsley}}{{r|Bityutskov|group=sec}}，或是[[概率论]]中的[[几乎必然]]{{r|Billingsley}}{{r|Ito2|group=sec}}。

例子：
* 在[[测度空间]]（例如實數）裡，可數集是零测集。[[有理数]]的集合可數，因此幾乎所有的實數都是無理數{{r|Niven}}。
* {{link-en|康托爾的第一篇集合論論文|Cantor's first set theory article}}證明了[[代數數]]的集合也是可數的，因此幾乎所有的實數都是[[超越數]]{{r|Baker}}{{r|group=sec|RealTrans}}。
* 幾乎所有實數都是[[正规数]]{{r|Granville}}。
* [[康托尔集]]也是零测集，雖然康托尔集不可數，但幾乎所有實數都不在內{{r|Korevaar}}。
* [[康托尔函数]]的導數在[[单位区间]]內幾乎所有數字下均為0{{r|Burk}}。這是以上範例的結果，因為康托尔函数是{{link-en|局部常數函數|locally constant functio}}，在康托尔集以下，其導數為0。

===數論中的意思===
{{further|漸進幾乎必然}}
[[数论]]中的「幾乎所有正整數」可以指「[[自然密度]]為1集合裡的正整數」。也就是說，若<var>A</var>是一個正整數的集合，當<var>n</var>趨近無限大時，小於<var>n</var>，在集合''A''裡的正整數數量，除以小於<var>n</var>的正整數數量，比值趨近於1，則幾乎所有整數都是在集合<var>A</var>內{{r|Hardy1|Hardy2}}{{r|Weisstein|group=sec}}。

若再進一步推廣，令<var>S</var>是正整數的無窮集合，例如正的偶數集合或是[[质数]]集合，若<var>A</var>是<var>S</var>的子集合，且當<var>n</var>趨近無限大時，若集合<var>A</var>裡小於<var>n</var>的元素數量，除以集合<var>S</var>裡小於<var>n</var>的元素數量，比值趨近於1，則可以說幾乎所有集合<var>S</var>裡的元素都在集合<var>A</var>裡。

例子：
* 正整數的{{link-en|餘有限集|cofinite set}}其自然密度為1，因此每一個餘有限集都包括幾乎所有的正整數。
* 幾乎所有正整數都是[[合数]]{{r|Weisstein|group=sec}}{{refn |group=proof |[[質數定理]]指出小於等於<var>n</var>的質數個數漸近等於<var>n</var>/ln(<var>n</var>)。因此，質數比例大約是ln(<var>n</var>)/<var>n</var>，隨著<var>n</var>趨近於[[无穷大]]，質數比例會趨近於0，而合數比例會趨近於1{{r|Hardy2}}}}
* 幾乎所有正的偶數都可以表示為二個質數的和{{r|Courant|page=489}}。
* 幾乎所有質數都不是[[孪生素数]]。進一步說，針對每一個正整數{{mvar|g}}，幾乎所有質數的[[質數間隙|間隙]]都大於{{mvar|g}}，幾乎所有質數和其較大質數以及較小質數的間隔都都大於{{mvar|g}}，也就是說，在{{math|''p'' − ''g''}}和{{math|''p'' + ''g''}}之間沒有其他的質數{{r|Prachar}}。
===拓撲學中的意思===
在[[拓扑学|topology]]{{r|Oxtoby}}，特別是[[动力系统理论]]中{{r|Baratchart|Broer|Sharkovsky}}（包括經濟學的應用）{{r|Yuan}} ，[[拓扑空间]]內幾乎所有的點可以指「除了在某個{{le|貧乏集|meagre set}}以外，所有此空間內的點。」有些則用更限定的定義，子集包括空間內幾乎所有的點，若這個子集包括某個[[开集]]的[[稠密集]]{{r|Broer|Albertini|Fuente}}。

例子：
* 給定某個{{link-en|超連通空間|hyperconnected space|不可約}}[[代数簇]]，在簇內幾乎所有點都符合的[[性质 (数学)|性质]]就是{{link-en|普遍性質|generic property}}{{r|Ito1|group=sec}}。這是因為在具有[[扎里斯基拓扑]]的不可約代数簇裡，所有非空的開集合都是稠密集

===代數中的意思===
在[[抽象代数]]和[[数理逻辑]]中，若<var>U</var>是集合<var>X</var>的[[超滤子]]，「集合<var>X</var>內幾乎所有元素」有時是指「<var>U</var>的部份元素內的元素」{{r|Komjath|Salzmann|Schoutens|Rautenberg}}。針對任何將<var>X</var>分為二個[[不交集]]的[[集合划分]]，其中一個不交集包括<var>X</var>裡幾乎所有的元素。<!--It is possible to think of the elements of a {{link-en||Filter (set theory)|filter}} on <var>X</var> as containing almost all elements of <var>X</var>, even if it isn't an ultrafilter.-->{{r|Rautenberg}}

==證明==
{{reflist|group=proof}}
==相關條目==
*{{link-en|足夠大|Sufficiently large}}
*[[幾乎]]
*[[幾乎處處]]
*[[几乎必然]]
==參考資料==
===一次文獻===
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<ref name=Cahen2>{{cite book |last1=Cahen |first1=Paul-Jean |last2=Chabert |first2=Jean-Luc |editor-last=Hazewinkel |editor-first=Michiel |date=7 December 2010 |orig-year=First published 2000 |title=Non-Noetherian Commutative Ring Theory |series=Mathematics and Its Applications |volume=520 |publisher=Springer |page=85 |chapter=Chapter 4: What's New About Integer-Valued Polynomials on a Subset? |doi=10.1007/978-1-4757-3180-4 |isbn=978-1-4419-4835-9}}</ref>
<ref name=Gardenfors>{{cite book |last=Gärdenfors |first=Peter  |date=22 August 2005 |title=The Dynamics of Thought |series=Synthese Library |volume=300 |publisher=Springer |pages=190–191 |isbn=978-1-4020-3398-8}}</ref>
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<ref name=Korevaar>{{cite book |last=Korevaar |first=Jacob |date=1 January 1968 |title=Mathematical Methods: Linear Algebra / Normed Spaces / Distributions / Integration |volume=1 |place=New York |publisher=Academic Press |pages=359–360 |isbn=978-1-4832-2813-6}}</ref>
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<ref name=Helmberg>{{cite book |last=Helmberg |first=Gilbert |date=December 1969 |title=Introduction to Spectral Theory in Hilbert Space |series=North-Holland Series in Applied Mathematics and Mechanics |volume=6 |edition=1st|place=Amsterdam |publisher=North-Holland Publishing Company |page=320 |isbn=978-0-7204-2356-3}}</ref>
<ref name=Vestrup>{{cite book |last=Vestrup |first=Eric M. |date=18 September 2003 |title=The Theory of Measures and Integration |url=https://archive.org/details/theoryofmeasures0000vest |series=Wiley Series in Probability and Statistics |place=United States |publisher=Wiley-Interscience |page=[https://archive.org/details/theoryofmeasures0000vest/page/n203 182] |isbn=978-0-471-24977-1}}</ref>
<ref name=Billingsley>{{cite book |last=Billingsley |first=Patrick |date=1 May 1995 |title=Probability and Measure |url=https://www.colorado.edu/amath/sites/default/files/attached-files/billingsley.pdf |series=Wiley Series in Probability and Statistics |edition=3rd |place=United States |publisher=Wiley-Interscience |page=60 |isbn=978-0-471-00710-4 |archive-url=https://web.archive.org/web/20180523011143/https://www.colorado.edu/amath/sites/default/files/attached-files/billingsley.pdf |archive-date=23 May 2018}}</ref>
<ref name=Niven>{{cite book |last=Niven |first=Ivan  |date=1 June 1956 |title=Irrational Numbers |series=Carus Mathematical Monographs |volume=11 |place=Rahway |publisher=Mathematical Association of America |pages=2–5 |isbn=978-0-88385-011-4}}</ref>
<ref name=Baker>{{cite book |last=Baker |first=Alan |date=1984 |title=A concise introduction to the theory of numbers |url=https://archive.org/details/conciseintroduct0000bake/page/53 |publisher=Cambridge University Press |page=[https://archive.org/details/conciseintroduct0000bake/page/53 53] |isbn=978-0-521-24383-4 }}</ref>
<ref name=Granville>{{cite book |last1=Granville |first1=Andrew  |last2=Rudnick |first2=Zeev  |date=7 January 2007 |title=Equidistribution in Number Theory, An Introduction |series=Nato Science Series II |volume=237 |publisher=Springer |page=11 |isbn=978-1-4020-5404-4}}</ref>
<ref name=Burk>{{cite book |last=Burk |first=Frank |date=3 November 1997 |title=Lebesgue Measure and Integration: An Introduction |series=A Wiley-Interscience Series of Texts, Monographs, and Tracts |place=United States |publisher=Wiley-Interscience |page=260 |isbn=978-0-471-17978-8}}</ref>
<ref name=Hardy1>{{cite book |last=Hardy |first=G. H.  |date=1940 |title=Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work |url=https://archive.org/details/pli.kerala.rare.37877 |publisher=Cambridge University Press |page=50}}</ref>
<ref name=Hardy2>{{cite book |last1=Hardy |first1=G. H. |last2=Wright |first2=E. M. |date=December 1960 |title=An Introduction to the Theory of Numbers |url=https://archive.org/details/AnIntroductionToTheTheoryOfNumbers-4thEd-G.h.HardyE.m.Wright |edition=4th |publisher=Oxford University Press |pages=8–9 |isbn=978-0-19-853310-8}}</ref>
<ref name=Prachar>{{cite book |last=Prachar |first=Karl |date=1957 |title=Primzahlverteilung |series=Grundlehren der mathematischen Wissenschaften |language=de |volume=91 |place=Berlin |publisher=Springer |page=164}} Cited in {{cite book |last=Grosswald |first=Emil |date=1 January 1984 |title=Topics from the Theory of Numbers |url=https://archive.org/details/topicsfromtheory0000emil |edition=2nd |place=Boston |publisher=Birkhäuser |page=[https://archive.org/details/topicsfromtheory0000emil/page/n49 30] |isbn=978-0-8176-3044-7}}</ref>
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<ref name=Babai>{{cite book |last=Babai |first=László |editor1-last=Graham |editor1-first=Ronald  |editor2-last=Grötschel |editor2-first=Martin  |editor3-last=Lovász |editor3-first=László |date=25 December 1995 |title=Handbook of Combinatorics |volume=2 |publisher=North-Holland Publishing Company |place=Netherlands |page=1462 |chapter=Automorphism Groups, Isomorphism, Reconstruction |isbn=978-0-444-82351-9}}</ref>
<ref name=Spencer>{{cite book |last=Spencer |first=Joel  |date=9 August 2001 |title=The Strange Logic of Random Graphs  |series=Algorithms and Combinatorics |volume=22 |publisher=Springer |pages=3–4 |isbn=978-3-540-41654-8}}</ref>
<ref name=Gradel>{{cite book |last1=Grädel |first1=Eric |last2=Kolaitis |first2=Phokion G. |last3=Libkin |first3=Leonid  |last4=Marx |first4=Maarten |last5=Spencer |first5=Joel |last6=Vardi |first6=Moshe Y.  |last7=Venema |first7=Yde |last8=Weinstein |first8=Scott |date=11 June 2007 |title=Finite Model Theory and Its Applications |url=https://archive.org/details/isbn_9783540004288 |series=Texts in Theoretical Computer Science (An EATCS Series) |publisher=Springer |page=[https://archive.org/details/isbn_9783540004288/page/n315 298] |isbn=978-3-540-00428-8}}</ref>
<ref name=Buckley>{{cite book |last1=Buckley |first1=Fred |last2=Harary |first2=Frank |date=21 January 1990 |title=Distance in Graphs |url=https://archive.org/details/distanceingraphs0000buck |publisher=Addison-Wesley |page=[https://archive.org/details/distanceingraphs0000buck/page/n126 109] |isbn=978-0-201-09591-3}}</ref>-->
<ref name=Oxtoby>{{cite book |last=Oxtoby |first=John C. |date=1980 |title=Measure and Category |series=Graduate Texts in Mathematics |volume=2 |edition=2nd |publisher=Springer |place=United States |pages=59,68 |isbn=978-0-387-90508-2}} While Oxtoby does not explicitly define the term there, Babai has borrowed it from ''Measure and Category'' in his chapter "Automorphism Groups, Isomorphism, Reconstruction" of Graham, Grötschel and Lovász's ''Handbook of Combinatorics'' (vol. 2), and Broer and Takens note in their book ''Dynamical Systems and Chaos'' that ''Measure and Category'' compares this meaning of "almost all" to the measure theoretic one in the real line (though Oxtoby's book discusses meagre sets in general topological spaces as well).</ref>
<ref name=Baratchart>{{cite book |last=Baratchart |first=Laurent |editor-last=Curtain |editor-first=Ruth F. |date=1987 |title=Modelling, Robustness and Sensitivity Reduction in Control Systems |series=NATO ASI Series F |volume=34 |publisher=[[Springer Science+Business Media|Springer]] |page=123 |chapter=Recent and New Results in Rational L<sup>2</sup> Approximation |doi=10.1007/978-3-642-87516-8 |isbn=978-3-642-87516-8}}</ref>
<ref name=Broer>{{cite book |last1=Broer |first1=Henk |last2=Takens |first2=Floris  |date=28 October 2010 |title=Dynamical Systems and Chaos |series=Applied Mathematical Sciences |volume=172 |publisher=[[Springer Science+Business Media|Springer]] |page=245 |doi=10.1007/978-1-4419-6870-8 |isbn=978-1-4419-6870-8}}</ref>
<ref name=Sharkovsky>{{cite book |last1=Sharkovsky |first1=A. N. |last2=Kolyada |first2=S. F. |last3=Sivak |first3=A. G. |last4=Fedorenko |first4=V. V. |date=30 April 1997 |title=Dynamics of One-Dimensional Maps |series=Mathematics and Its Applications |volume=407 |publisher=[[Springer Science+Business Media|Springer]] |page=33 |doi=10.1007/978-94-015-8897-3 |isbn=978-94-015-8897-3}}</ref>
<ref name=Yuan>{{cite book |last=Yuan |first=George Xian-Zhi |date=9 February 1999 |title=KKM Theory and Applications in Nonlinear Analysis |series=Pure and Applied Mathematics; A Series of Monographs and Textbooks |publisher=Marcel Dekker |page=21 |isbn=978-0-8247-0031-7}}</ref>
<ref name=Albertini>{{cite book |last1=Albertini |first1=Francesca |last2=Sontag |first2=Eduardo D. |editor1-last=Bonnard |editor1-first=Bernard |editor2-last=Bride |editor2-first=Bernard |editor3-last=Gauthier |editor3-first=Jean-Paul |editor4-last=Kupka |editor4-first=Ivan |date=1 September 1991 |title=Analysis of Controlled Dynamical Systems |series=Progress in Systems and Control Theory |volume=8 |publisher=Birkhäuser |page=29 |chapter=Transitivity and Forward Accessibility of Discrete-Time Nonlinear Systems |doi=10.1007/978-1-4612-3214-8 |isbn=978-1-4612-3214-8}}</ref>
<ref name=Fuente>{{cite book |last=De la Fuente |first=Angel |date=28 January 2000 |title=Mathematical Models and Methods for Economists |publisher=[[Cambridge University Press]] |page=217 |isbn=978-0-521-58529-3}}</ref>
<ref name=Komjath>{{cite book |last1=Komjáth |first1=Péter |last2=Totik |first2=Vilmos |date=2 May 2006 |title=Problems and Theorems in Classical Set Theory |series=Problem Books in Mathematics |publisher=[[Springer Science+Business Media|Springer]] |place=United States |page=75 |isbn=978-0387-30293-5}}</ref>
<ref name=Salzmann>{{cite book |last1=Salzmann |first1=Helmut |last2=Grundhöfer |first2=Theo |last3=Hähl |first3=Hermann |last4=Löwen |first4=Rainer |date=24 September 2007 |title=The Classical Fields: Structural Features of the Real and Rational Numbers |series=Encyclopedia of Mathematics and Its Applications |volume=112 |publisher=[[Cambridge University Press]] |page=[https://archive.org/details/classicalfieldss0000unse/page/155 155] |isbn=978-0-521-86516-6 |url=https://archive.org/details/classicalfieldss0000unse/page/155 }}</ref>
<ref name=Schoutens>{{cite book |last=Schoutens|first=Hans |date=2 August 2010 |title=The Use of Ultraproducts in Commutative Algebra |series=Lecture Notes in Mathematics |volume=1999 |publisher=[[Springer Science+Business Media|Springer]] |page=8 |doi=10.1007/978-3-642-13368-8 |isbn=978-3-642-13367-1}}</ref>
<ref name=Rautenberg>{{cite book |last=Rautenberg |first=Wolfgang |date=17 December 2009 |title=A Concise to Mathematical Logic |series=Universitext |edition=3rd |publisher=[[Springer Science+Business Media|Springer]] |pages=210–212 |doi=10.1007/978-1-4419-1221-3 |isbn=978-1-4419-1221-3}}</ref>
}}
===二次文獻===
{{reflist |group=sec |refs=
<ref name=RealTrans>{{Cite web|url=https://proofwiki.org/wiki/Almost_All_Real_Numbers_are_Transcendental|title=Almost All Real Numbers are Transcendental - ProofWiki|website=proofwiki.org|access-date=2019-11-11}}</ref>
<ref name=Schwartzman>{{cite book |last=Schwartzman |first=Steven |date=1 May 1994 |title=The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English |url=https://archive.org/details/wordsofmathemati0000schw |url-access=registration |series=Spectrum Series |publisher=[[Mathematical Association of America]] |page=[https://archive.org/details/wordsofmathemati0000schw/page/22 22] |isbn=978-0-88385-511-9}}</ref>
<ref name=Clapham>{{cite book |last1=Clapham |first1=Christopher |last2=Nicholson |first2=James |date=7 June 2009 |title=The Concise Oxford Dictionary of mathematics |series=Oxford Paperback References |edition=4th |page=38 |publisher=[[Oxford University Press]] |isbn=978-0-19-923594-0}}</ref>
<ref name=James>{{cite book |last=James |first=Robert C. |date=31 July 1992 |title=Mathematics Dictionary |edition=5th |publisher=Chapman & Hall |page=269 |isbn=978-0-412-99031-1}}</ref>
<ref name=Bityutskov>{{cite book |last=Bityutskov |first=Vadim I. |editor-last=Hazewinkel |editor-first=Michiel  |date=30 November 1987 |title=Encyclopaedia of Mathematics |volume=1 |publisher=[[Kluwer Academic Publishers]] |page=153 |chapter=Almost-everywhere |chapter-url=http://www.encyclopediaofmath.org/index.php?title=Almost-everywhere&oldid=31533 |doi=10.1007/978-94-015-1239-8 |isbn=978-94-015-1239-8}}</ref>
<ref name=Ito2>{{cite book |editor-last=Itô |editor-first=Kiyosi |date=4 June 1993 |title=Encyclopedic Dictionary of Mathematics |edition=2nd |volume=2 |publisher=[[MIT Press]] |place=Kingsport |page=1267 |isbn=978-0-262-09026-1}}</ref>
<ref name=Weisstein>{{MathWorld|title=Almost All|urlname=AlmostAll}} See also {{cite book |last=Weisstein |first=Eric W.  |date=25 November 1988 |title=CRC Concise Encyclopedia of Mathematics |url=https://archive.org/details/CrcEncyclopediaOfMathematics |edition=1st |publisher=[[CRC Press]] |page=41 |isbn=978-0-8493-9640-3}}</ref>
<ref name=Ito1>{{cite book |editor-last=Itô |editor-first=Kiyosi |date=4 June 1993 |title=Encyclopedic Dictionary of Mathematics |url=https://archive.org/stream/Ito_Kiyoso_-_Encyclopedic_Dictionary_Of_Math_Volume_1#page/n85/mode/2up |edition=2nd |volume=1 |publisher=[[MIT Press]] |place=Kingsport |page=67 |isbn=978-0-262-09026-1}}</ref>
}}

==外部連結==
*{{MathWorld|title=Almost All|urlname=AlmostAll}}

[[Category:数学术语]]
[[Category:數學表示法]]