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In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if is a set and is a subset of whose complement in is negligible, then almost all elements of are in . The meaning of "negligible" depends on the mathematical context: for instance, it can mean finite, null, or meagre.

Contents

Prevalent meaningEdit

Across mathematics, when referring to a subset of an infinite set, "almost all" is sometimes used to mean "all but finitely many".[1][2][3] This use occurs in philosophy as well.[4]

Examples:

  • Almost all positive integers are greater than 1,000,000,000,000.[5]:293
  • Almost all prime numbers are odd, as 2 is the only exception.
  • Almost all polyhedra are irregular, as there are only nine exceptions: the five platonic solids and the four Kepler-Poinsot polyhedra.
  • If   and   are coprime positive integers, almost all positive integers can be expressed as   where   and   are positive integers.
  • If   is a nonzero polynomial,   for almost all  .

Meaning in measure theoryEdit

 
The Cantor function

When speaking about the reals, sometimes "almost all" means "all reals but a null set". Similarly, if   is some set of real numbers, "almost all numbers in  " can mean "all numbers in   but those in a null set".[6][7][8]

The real line can be thought of as a one-dimensional Euclidean space. In the more general case of an  -dimensional space (where   is a positive integer), the definition can be generalised to "all points in   but those in a null set" (this time,   is a set of points in the space).[9][7]

Even more generally, "almost all" is sometimes used in the sense of "almost everywhere" in measure theory,[10][11] or in the closely related sense of "almost surely" in probability theory.[11]

Examples:

Meaning in number theoryEdit

In number theory, "almost all positive integers" can mean "a set of positive integers whose natural density is 1". That is, if   is a set of positive integers, and if the proportion of positive integers below   that are in   tends to 1 as   tends to infinity (see limit), then almost all positive integers are in  .[16][17][18]

More generally, let   be an infinite set of positive integers, such as the set of even positive numbers or of prime numbers. If   is a subset of  , and if the proportion of elements of   below   that are in   tends to 1 as   tends to infinity, then it can be said that almost all elements of   are in  .

Examples:

  • Almost all positive integers are composite. [proof 1]
  • Almost all even positive numbers can be expressed as the sum of two primes.[5]:489
  • Almost all primes are isolated. Moreover, if   is an arbitrary positive integer, then almost all primes have prime gaps of more than   both to their left and to their right (no other primes between   and  .[19]

Meaning in topologyEdit

In general topology, subset of a topological space contains "almost all" of its points if the subset is open and dense, or more generally, if it is comeagre.[citation needed] In the latter case, it can be said to have "Baire-almost all" of the space's points. This term is used, for instance, in the context of certain topological spaces that arise in set theory,[20] and certain function spaces in functional analysis.[21]

Meaning in algebraEdit

In abstract algebra, if   is an ultrafilter on a set  , "almost all elements of  " means "the elements of some element of  ".[22][23] If   is divided into two disjoint sets, no matter how, one of them will contain almost all elements of  .

GeneralisationsEdit

ContentsEdit

Given a set   and a content on it, a subset of   can be said to contain "almost all elements of  " if the content of its complement is 0 (the complement is null). This is a generalisation of the meanings in measure theory (all measures are contents), number theory (natural density is a content), and algebra (if the content of a set in the ultrafilter is defined to be 1 and the content of a set not in the ultrafilter is defined to be 0).

FiltersEdit

Given a set   and a filter on it, a subset of   can be said to contain "almost all elements of  " if it is an element of the filter. This is a generalisation of all meanings illustrated here (the filter comprises the subsets that contain "almost all" elements of  ).

See alsoEdit

ProofsEdit

  1. ^ According to the prime number theorem, the number of prime numbers less than or equal to   is asymptotically equal to  . Therefore, the proportion of primes is roughly  , which tends to 0 as   tends to infinity, so the proportion of composite numbers less than or equal to   tends to 1 as   tends to infinity.[18]

ReferencesEdit

  1. ^ Cahen, Paul-Jean; Chabert, Jean-Luc (3 December 1996). Integer-Valued Polynomials. Mathematical Surveys and Monographs. 48. American Mathematical Society. p. xix. ISBN 978-0-8218-0388-2. ISSN 0076-5376. 
  2. ^ Cahen, Paul-Jean; Chabert, Jean-Luc (7 December 2010) [First published 2000]. "Chapter 4: What's New About Integer-Valued Polynomials on a Subset?". In Hazewinkel, Michiel. Non-Noetherian Commutative Ring Theory. Mathematics and Its Applications. 520. Dordrecht: Springer. p. 85. doi:10.1007/978-1-4757-3180-4. ISBN 978-1-4419-4835-9. 
  3. ^ Halmos, Paul R. (1962). Algebric Logic. New York: Chelsea Publishing Company. p. 114. 
  4. ^ Gärdenfors, Peter (22 August 2005). The Dynamics of Thought. Synthese Library. 300. Dordrecht: Springer. pp. 190–191. ISBN 978-1-4020-3398-8. 
  5. ^ a b Courant, Richard; Robbins, Herbert; Stewart, Ian (18 July 1996). What is Mathematics? An Elementary Approach to Ideas and Methods (PDF) (2nd ed.). Oxford University Press. ISBN 978-0-19-510519-3. 
  6. ^ a b Korevaar, Jacob (1 January 1968). Mathematical Methods: Linear Algebra / Normed Spaces / Distributions / Integration. 1. New York: Academic Press. pp. 359–360. ISBN 978-1-4832-2813-6. 
  7. ^ a b Natanson, Isidor P. (1955). Theory of Functions of a Real Variable. 1. Translated by Boron, Leo F. (revised ed.). New York: Frederick Ungar Publishing (published 1983). p. 90. ISBN 978-0-8044-7020-9. 
  8. ^ Leine, Remco I.; van de Wouw, Nathan (4 March 2008). Stability and Convergence of Mechanical Systems with Unilateral Constraints. Lecture Notes in Applied and Computational Mechanics. 36. Springer. p. 63. ISBN 978-3-540-76974-3. 
  9. ^ Helmberg, Gilbert (December 1969). Introduction to Spectral Theory in Hilbert Space. North-Holland Series in Applied Mathematics and Mechanics. 6 (1st ed.). Amsterdam: North-Holland Publishing Company (published 1975). p. 320. ISBN 978-0-7204-2356-3. 
  10. ^ Vestrup, Eric M. (18 September 2003). The Theory of Measures and Integration. Wiley Series in Probability and Statistics. United States: Wiley-Interscience. p. 182. ISBN 978-0-471-24977-1. 
  11. ^ a b Billingsley, Patrick (1 May 1995). Probability and Measure (PDF). Wiley Series in Probability and Statistics (3rd ed.). United States: Wiley-Interscience. p. 60. ISBN 978-0-471-00710-4. Archived from the original (PDF) on 23 May 2018. 
  12. ^ Niven, Ivan (1 June 1956). Irrational Numbers. Carus Mathematical Monographs. 11. Rahway: Mathematical Association of America. pp. 2–5. ISBN 978-0-88385-011-4. 
  13. ^ Baker, Alan (1984). A concise introduction to the theory of numbers. Cambridge University Press. p. 53. ISBN 978-0-521-24383-4. 
  14. ^ Granville, Andrew; Rudnick, Zeev (7 January 2007). Equidistribution in Number Theory, An Introduction. Nato Science Series II. 237. Dordrecht: Springer. p. 11. ISBN 978-1-4020-5404-4. 
  15. ^ Burk, Frank (3 November 1997). Lebesgue Measure and Integration: An Introduction. A Wiley-Interscience Series of Texts, Monographs, and Tracts. United States: Wiley-Interscience. p. 260. ISBN 978-0-471-17978-8. 
  16. ^ Weisstein, Eric W. "Almost All". MathWorld. 
  17. ^ Hardy, G. H. (1940). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Cambridge University Press. p. 50. 
  18. ^ a b Hardy, G. H.; Wright, E. M. (December 1960). Introduction to the Theory of Numbers (4th ed.). Oxford University Press (published 1975). pp. 8–9. ISBN 978-0-19-853310-8. 
  19. ^ Prachar, Karl (1957). Primzahlverteilung. Grundlehren der mathematischen Wissenschaften (in German). 91. Berlin: Springer. p. 164.  Cited in Grosswald, Emil (1 January 1984). Topics from the Theory of Numbers (2nd ed.). Boston: Birkhäuser. p. 30. ISBN 978-0-8176-3044-7. 
  20. ^ Hinman, Peter G. (11 July 1978). Recursion-Theoretic Hierarchies. Perspectives in Mathematical Logic. Würzburg: Springer. p. 220. ISBN 978-3-540-07904-0. 
  21. ^ Brown, B. Malcolm; Eastham, Michael S. P.; Schmidt, Karl Michael (30 October 2012). Periodic Differential Operators. Operator Theory: Advances and Applications. 230. Basel: Birkhäuser. p. 105. doi:10.1007/978-3-0348-052. ISBN 978-3-0348-0527-8. 
  22. ^ Salzmann, Helmut; Grundhöfer, Theo; Hähl, Hermann; Löwen, Rainer (24 September 2007). The Classical Fields: Structural Features of the Real and Rational Numbers. Encyclopedia of Mathematics and Its Applications. 112. Cambridge University Press. p. 155. ISBN 978-0-521-86516-6. 
  23. ^ Schoutens, Hans (2 August 2010). The Use of Ultraproducts in Commutative Algebra. Lecture Notes in Mathematics. 1999. Springer. p. 8. doi:10.1007/978-3-642-13368-8. ISBN 978-3-642-13367-1.