Almost all

Inmathematics,the term "almost all"means" all but a negligible quantity ". More precisely, if $X$ is aset,"almost all elements of $X$ "means" all elements of $X$ but those in anegligible subsetof $X$ ".The meaning of" negligible "depends on the mathematical context; for instance, it can meanfinite,countable,ornull.

In contrast, "almost no"means" a negligible quantity "; that is," almost no elements of $X$ "means" a negligible quantity of elements of $X$ ".

Meanings in different areas of mathematics

Prevalent meaning

Throughout mathematics, "almost all" is sometimes used to mean "all (elements of aninfinite set) except forfinitelymany ".^[1]^[2]This use occurs in philosophy as well.^[3]Similarly, "almost all" can mean "all (elements of anuncountable set) except forcountablymany ".^{[sec 1]}

Examples:

Almost all positive integers are greater than 10¹².^[4]^: 293
Almost allprime numbersare odd (2 is the only exception).^[5]
Almost allpolyhedraareirregular(as there are only nine exceptions: the fiveplatonic solidsand the fourKepler–Poinsot polyhedra).
IfPis a nonzeropolynomial,thenP(x)≠ 0 for almost allx(if not allx).

Meaning in measure theory

TheCantor functionas a function that has zero derivative almost everywhere

When speaking about thereals,sometimes "almost all" can mean "all reals except for anull set".^[6]^[7]^{[sec 2]}Similarly, ifSis some set of reals, "almost all numbers inS"can mean" all numbers inSexcept for those in a null set ".^[8]Thereal linecan be thought of as a one-dimensionalEuclidean space.In the more general case of ann-dimensional space (wherenis a positive integer), these definitions can begeneralisedto "all points except for those in a null set"^{[sec 3]}or "all points inSexcept for those in a null set "(this time,Sis a set of points in the space).^[9]Even more generally, "almost all" is sometimes used in the sense of "almost everywhere"inmeasure theory,^[10]^[11]^{[sec 4]}or in the closely related sense of "almost surely"inprobability theory.^[11]^{[sec 5]}

Examples:

In ameasure space,such as the real line, countable sets are null. The set ofrational numbersis countable, so almost all real numbers are irrational.^[12]
GeorgCantor's first set theory articleproved that the set ofalgebraic numbersis countable as well, so almost all reals aretranscendental.^[13]^{[sec 6]}
Almost all reals arenormal.^[14]
TheCantor setis also null. Thus, almost all reals are not in it even though it is uncountable.^[6]
The derivative of theCantor functionis 0 for almost all numbers in theunit interval.^[15]It follows from the previous example because the Cantor function islocally constant,and thus has derivative 0 outside the Cantor set.

Meaning in number theory

Innumber theory,"almost all positive integers" can mean "the positive integers in a set whosenatural densityis 1 ". That is, ifAis a set of positive integers, and if the proportion of positive integers inAbelown(out of all positive integers belown)tends to1 asntends to infinity, then almost all positive integers are inA.^[16]^[17]^{[sec 7]}

More generally, letSbe an infinite set of positive integers, such as the set of even positive numbers or the set ofprimes,ifAis a subset ofS,and if the proportion of elements ofSbelownthat are inA(out of all elements ofSbelown) tends to 1 asntends to infinity, then it can be said that almost all elements ofSare inA.

Examples:

The natural density ofcofinite setsof positive integers is 1, so each of them contains almost all positive integers.
Almost all positive integers arecomposite.^{[sec 7]}^{[proof 1]}
Almost all even positive numbers can be expressed as the sum of two primes.^[4]^: 489
Almost all primes areisolated.Moreover, for every positive integer $g$ ,almost all primes haveprime gapsof more than $g$ both to their left and to their right; that is, there is no other prime between $p - g$ and $p + g$ .^[18]

Meaning in graph theory

Ingraph theory,ifAis a set of (finitelabelled)graphs,it can be said to contain almost all graphs, if the proportion of graphs withnvertices that are inAtends to 1 asntends to infinity.^[19]However, it is sometimes easier to work with probabilities,^[20]so the definition is reformulated as follows. The proportion of graphs withnvertices that are inAequals the probability that a random graph withnvertices (chosen with theuniform distribution) is inA,and choosing a graph in this way has the same outcome as generating a graph by flipping a coin for each pair of vertices to decide whether to connect them.^[21]Therefore, equivalently to the preceding definition, the setAcontains almost all graphs if the probability that a coin-flip–generated graph withnvertices is inAtends to 1 asntends to infinity.^[20]^[22]Sometimes, the latter definition is modified so that the graph is chosen randomly in someother way,where not all graphs withnvertices have the same probability,^[21]and those modified definitions are not always equivalent to the main one.

The use of the term "almost all" in graph theory is not standard; the term "asymptotically almost surely"is more commonly used for this concept.^[20]

Example:

Almost all graphs areasymmetric.^[19]
Almost all graphs havediameter2.^[23]

Meaning in topology

Intopology^[24]and especiallydynamical systems theory^[25]^[26]^[27](including applications in economics),^[28]"almost all" of atopological space's points can mean "all of the space's points except for those in ameagre set".Some use a more limited definition, where a subset contains almost all of the space's points only if it contains someopen dense set.^[26]^[29]^[30]

Example:

Given anirreducible algebraic variety,thepropertiesthat hold for almost all points in the variety are exactly thegeneric properties.^{[sec 8]}This is due to the fact that in an irreducible algebraic variety equipped with theZariski topology,all nonempty open sets are dense.

Meaning in algebra

Inabstract algebraandmathematical logic,ifUis anultrafilteron a setX,"almost all elements ofX"sometimes means" the elements of someelementofU".^[31]^[32]^[33]^[34]For anypartitionofXinto twodisjoint sets,one of them will necessarily contain almost all elements ofX.It is possible to think of the elements of afilteronXas containing almost all elements ofX,even if it isn't an ultrafilter.^[34]

Proofs

^Theprime number theoremshows that the number of primes less than or equal tonis asymptotically equal ton/ln(n). Therefore, the proportion of primes is roughly ln(n)/n,which tends to 0 asntends toinfinity,so the proportion of composite numbers less than or equal tontends to 1 asntends to infinity.^[17]

References

Primary sources

^Cahen, Paul-Jean; Chabert, Jean-Luc (3 December 1996).Integer-Valued Polynomials.Mathematical Surveys and Monographs.Vol. 48.American Mathematical Society.p. xix.ISBN 978-0-8218-0388-2.ISSN 0076-5376.
^Cahen, Paul-Jean; Chabert, Jean-Luc (7 December 2010) [First published 2000]. "Chapter 4: What's New About Integer-Valued Polynomials on a Subset?". InHazewinkel, Michiel(ed.).Non-Noetherian Commutative Ring Theory.Mathematics and Its Applications. Vol. 520.Springer.p. 85.doi:10.1007/978-1-4757-3180-4.ISBN 978-1-4419-4835-9.
^Gärdenfors, Peter(22 August 2005).The Dynamics of Thought.Synthese Library. Vol. 300.Springer.pp. 190–191.ISBN 978-1-4020-3398-8.
^^a ^bCourant, Richard;Robbins, Herbert;Stewart, Ian(18 July 1996).What is Mathematics? An Elementary Approach to Ideas and Methods(2nd ed.).Oxford University Press.ISBN 978-0-19-510519-3.
^Movshovitz-hadar, Nitsa; Shriki, Atara (2018-10-08).Logic In Wonderland: An Introduction To Logic Through Reading Alice's Adventures In Wonderland - Teacher's Guidebook.World Scientific. p. 38.ISBN 978-981-320-864-3.This can also be expressed in the statement: 'Almost all prime numbers are odd.'
^^a ^bKorevaar, Jacob(1 January 1968).Mathematical Methods: Linear Algebra / Normed Spaces / Distributions / Integration.Vol. 1. New York:Academic Press.pp. 359–360.ISBN 978-1-4832-2813-6.
^Natanson, Isidor P.(June 1961).Theory of Functions of a Real Variable.Vol. 1. Translated by Boron, Leo F. (revised ed.). New York:Frederick Ungar Publishing.p. 90.ISBN 978-0-8044-7020-9.
^Sohrab, Houshang H. (15 November 2014).Basic Real Analysis(2 ed.).Birkhäuser.p. 307.doi:10.1007/978-1-4939-1841-6.ISBN 978-1-4939-1841-6.
^Helmberg, Gilbert (December 1969).Introduction to Spectral Theory in Hilbert Space.North-Holland Series in Applied Mathematics and Mechanics. Vol. 6 (1st ed.). Amsterdam:North-Holland Publishing Company.p. 320.ISBN 978-0-7204-2356-3.
^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.
^^a ^bBillingsley, 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 fromthe original(PDF)on 23 May 2018.
^Niven, Ivan(1 June 1956).Irrational Numbers.Carus Mathematical Monographs.Vol. 11. Rahway:Mathematical Association of America.pp. 2–5.ISBN 978-0-88385-011-4.
^Baker, Alan(1984).A concise introduction to the theory of numbers.Cambridge University Press.p.53.ISBN 978-0-521-24383-4.
^Granville, Andrew;Rudnick, Zeev(7 January 2007).Equidistribution in Number Theory, An Introduction.Nato Science Series II. Vol. 237.Springer.p. 11.ISBN 978-1-4020-5404-4.
^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.
^Hardy, G. H.(1940).Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work.Cambridge University Press.p. 50.
^^a ^bHardy, G. H.;Wright, E. M.(December 1960).An Introduction to the Theory of Numbers(4th ed.).Oxford University Press.pp. 8–9.ISBN 978-0-19-853310-8.
^Prachar, Karl(1957).Primzahlverteilung.Grundlehren der mathematischen Wissenschaften (in German). Vol. 91. Berlin:Springer.p. 164.Cited inGrosswald, Emil(1 January 1984).Topics from the Theory of Numbers(2nd ed.). Boston:Birkhäuser.p. 30.ISBN 978-0-8176-3044-7.
^^a ^bBabai, László(25 December 1995). "Automorphism Groups, Isomorphism, Reconstruction". InGraham, Ronald;Grötschel, Martin;Lovász, László(eds.).Handbook of Combinatorics.Vol. 2. Netherlands:North-Holland Publishing Company.p. 1462.ISBN 978-0-444-82351-9.
^^a ^b ^cSpencer, Joel(9 August 2001).The Strange Logic of Random Graphs.Algorithms and Combinatorics. Vol. 22.Springer.pp. 3–4.ISBN 978-3-540-41654-8.
^^a ^bBollobás, Béla(8 October 2001).Random Graphs.Cambridge Studies in Advanced Mathematics. Vol. 73 (2nd ed.).Cambridge University Press.pp. 34–36.ISBN 978-0-521-79722-1.
^Grädel, Eric; Kolaitis, Phokion G.;Libkin, Leonid;Marx, Maarten;Spencer, Joel;Vardi, Moshe Y.;Venema, Yde; Weinstein, Scott (11 June 2007).Finite Model Theory and Its Applications.Texts in Theoretical Computer Science (AnEATCSSeries).Springer.p. 298.ISBN 978-3-540-00428-8.
^Buckley, Fred;Harary, Frank(21 January 1990).Distance in Graphs.Addison-Wesley.p. 109.ISBN 978-0-201-09591-3.
^Oxtoby, John C. (1980).Measure and Category.Graduate Texts in Mathematics.Vol. 2 (2nd ed.). United States:Springer.pp. 59, 68.ISBN 978-0-387-90508-2.While Oxtoby does not explicitly define the term there,Babaihas borrowed it fromMeasure and Categoryin his chapter "Automorphism Groups, Isomorphism, Reconstruction" of Graham,GrötschelandLovász'sHandbook of Combinatorics(vol. 2), and Broer andTakensnote in their bookDynamical Systems and ChaosthatMeasure and Categorycompares 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).
^Baratchart, Laurent (1987). "Recent and New Results in Rational L²Approximation ". InCurtain, Ruth F.(ed.).Modelling, Robustness and Sensitivity Reduction in Control Systems.NATO ASI Series F. Vol. 34.Springer.p. 123.doi:10.1007/978-3-642-87516-8.ISBN 978-3-642-87516-8.
^^a ^bBroer, Henk;Takens, Floris(28 October 2010).Dynamical Systems and Chaos.Applied Mathematical Sciences. Vol. 172.Springer.p. 245.doi:10.1007/978-1-4419-6870-8.ISBN 978-1-4419-6870-8.
^Sharkovsky, A. N.; Kolyada, S. F.; Sivak, A. G.; Fedorenko, V. V. (30 April 1997).Dynamics of One-Dimensional Maps.Mathematics and Its Applications. Vol. 407.Springer.p. 33.doi:10.1007/978-94-015-8897-3.ISBN 978-94-015-8897-3.
^Yuan, George Xian-Zhi (9 February 1999).KKM Theory and Applications in Nonlinear Analysis.Pure and Applied Mathematics; A Series of Monographs and Textbooks.Marcel Dekker.p. 21.ISBN 978-0-8247-0031-7.
^Albertini, Francesca;Sontag, Eduardo D.(1 September 1991). "Transitivity and Forward Accessibility of Discrete-Time Nonlinear Systems". In Bonnard, Bernard; Bride, Bernard; Gauthier, Jean-Paul; Kupka, Ivan (eds.).Analysis of Controlled Dynamical Systems.Progress in Systems and Control Theory. Vol. 8.Birkhäuser.p. 29.doi:10.1007/978-1-4612-3214-8.ISBN 978-1-4612-3214-8.
^De la Fuente, Angel (28 January 2000).Mathematical Models and Methods for Economists.Cambridge University Press.p. 217.ISBN 978-0-521-58529-3.
^Komjáth, Péter;Totik, Vilmos(2 May 2006).Problems and Theorems in Classical Set Theory.Problem Books in Mathematics. United States:Springer.p. 75.ISBN 978-0387-30293-5.
^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. Vol. 112.Cambridge University Press.p.155.ISBN 978-0-521-86516-6.
^Schoutens, Hans (2 August 2010).The Use of Ultraproducts in Commutative Algebra.Lecture Notes in Mathematics.Vol. 1999.Springer.p. 8.doi:10.1007/978-3-642-13368-8.ISBN 978-3-642-13367-1.
^^a ^bRautenberg, Wolfgang(17 December 2009).A Concise to Mathematical Logic.Universitext (3rd ed.).Springer.pp. 210–212.doi:10.1007/978-1-4419-1221-3.ISBN 978-1-4419-1221-3.

Secondary sources

^Schwartzman, Steven (1 May 1994).The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English.Spectrum Series.Mathematical Association of America.p.22.ISBN 978-0-88385-511-9.
^Clapham, Christopher; Nicholson, James (7 June 2009).The Concise Oxford Dictionary of mathematics.Oxford Paperback References (4th ed.).Oxford University Press.p. 38.ISBN 978-0-19-923594-0.
^James, Robert C.(31 July 1992).Mathematics Dictionary(5th ed.).Chapman & Hall.p. 269.ISBN 978-0-412-99031-1.
^Bityutskov, Vadim I. (30 November 1987)."Almost-everywhere".InHazewinkel, Michiel(ed.).Encyclopaedia of Mathematics.Vol. 1.Kluwer Academic Publishers.p. 153.doi:10.1007/978-94-015-1239-8.ISBN 978-94-015-1239-8.
^Itô, Kiyosi,ed. (4 June 1993).Encyclopedic Dictionary of Mathematics.Vol. 2 (2nd ed.). Kingsport:MIT Press.p. 1267.ISBN 978-0-262-09026-1.
^"Almost All Real Numbers are Transcendental - ProofWiki".proofwiki.org.Retrieved2019-11-11.
^^a ^bWeisstein, Eric W."Almost All".MathWorld.See alsoWeisstein, Eric W.(25 November 1988).CRC Concise Encyclopedia of Mathematics(1st ed.).CRC Press.p. 41.ISBN 978-0-8493-9640-3.
^Itô, Kiyosi,ed. (4 June 1993).Encyclopedic Dictionary of Mathematics.Vol. 1 (2nd ed.). Kingsport:MIT Press.p. 67.ISBN 978-0-262-09026-1.

[25] Theprime number theoremshows that the number of primes less than or equal tonis asymptotically equal ton/ln(n). Therefore, the proportion of primes is roughly ln(n)/n,which tends to 0 asntends toinfinity,so the proportion of composite numbers less than or equal tontends to 1 asntends to infinity.^[17]

[Cahen1-1] Cahen, Paul-Jean; Chabert, Jean-Luc (3 December 1996).Integer-Valued Polynomials.Mathematical Surveys and Monographs.Vol. 48.American Mathematical Society.p. xix.ISBN 978-0-8218-0388-2.ISSN 0076-5376.

[Cahen2-2] Cahen, Paul-Jean; Chabert, Jean-Luc (7 December 2010) [First published 2000]. "Chapter 4: What's New About Integer-Valued Polynomials on a Subset?". InHazewinkel, Michiel(ed.).Non-Noetherian Commutative Ring Theory.Mathematics and Its Applications. Vol. 520.Springer.p. 85.doi:10.1007/978-1-4757-3180-4.ISBN 978-1-4419-4835-9.

[Gardenfors-3] Gärdenfors, Peter(22 August 2005).The Dynamics of Thought.Synthese Library. Vol. 300.Springer.pp. 190–191.ISBN 978-1-4020-3398-8.

[Courant-5] Courant, Richard;Robbins, Herbert;Stewart, Ian(18 July 1996).What is Mathematics? An Elementary Approach to Ideas and Methods(2nd ed.).Oxford University Press.ISBN 978-0-19-510519-3.

[6] Movshovitz-hadar, Nitsa; Shriki, Atara (2018-10-08).Logic In Wonderland: An Introduction To Logic Through Reading Alice's Adventures In Wonderland - Teacher's Guidebook.World Scientific. p. 38.ISBN 978-981-320-864-3.This can also be expressed in the statement: 'Almost all prime numbers are odd.'

[Korevaar-7] Korevaar, Jacob(1 January 1968).Mathematical Methods: Linear Algebra / Normed Spaces / Distributions / Integration.Vol. 1. New York:Academic Press.pp. 359–360.ISBN 978-1-4832-2813-6.

[Natanson-8] Natanson, Isidor P.(June 1961).Theory of Functions of a Real Variable.Vol. 1. Translated by Boron, Leo F. (revised ed.). New York:Frederick Ungar Publishing.p. 90.ISBN 978-0-8044-7020-9.

[Sohrab-10] Sohrab, Houshang H. (15 November 2014).Basic Real Analysis(2 ed.).Birkhäuser.p. 307.doi:10.1007/978-1-4939-1841-6.ISBN 978-1-4939-1841-6.

[Helmberg-12] Helmberg, Gilbert (December 1969).Introduction to Spectral Theory in Hilbert Space.North-Holland Series in Applied Mathematics and Mechanics. Vol. 6 (1st ed.). Amsterdam:North-Holland Publishing Company.p. 320.ISBN 978-0-7204-2356-3.

[Vestrup-13] 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.

[Billingsley-14] 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 fromthe original(PDF)on 23 May 2018.

[Niven-17] Niven, Ivan(1 June 1956).Irrational Numbers.Carus Mathematical Monographs.Vol. 11. Rahway:Mathematical Association of America.pp. 2–5.ISBN 978-0-88385-011-4.

[Baker-18] Baker, Alan(1984).A concise introduction to the theory of numbers.Cambridge University Press.p.53.ISBN 978-0-521-24383-4.

[Granville-20] Granville, Andrew;Rudnick, Zeev(7 January 2007).Equidistribution in Number Theory, An Introduction.Nato Science Series II. Vol. 237.Springer.p. 11.ISBN 978-1-4020-5404-4.

[Burk-21] 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.

[Hardy1-22] Hardy, G. H.(1940).Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work.Cambridge University Press.p. 50.

[Hardy2-23] Hardy, G. H.;Wright, E. M.(December 1960).An Introduction to the Theory of Numbers(4th ed.).Oxford University Press.pp. 8–9.ISBN 978-0-19-853310-8.

[Prachar-26] Prachar, Karl(1957).Primzahlverteilung.Grundlehren der mathematischen Wissenschaften (in German). Vol. 91. Berlin:Springer.p. 164.Cited inGrosswald, Emil(1 January 1984).Topics from the Theory of Numbers(2nd ed.). Boston:Birkhäuser.p. 30.ISBN 978-0-8176-3044-7.

[Babai-27] Babai, László(25 December 1995). "Automorphism Groups, Isomorphism, Reconstruction". InGraham, Ronald;Grötschel, Martin;Lovász, László(eds.).Handbook of Combinatorics.Vol. 2. Netherlands:North-Holland Publishing Company.p. 1462.ISBN 978-0-444-82351-9.

[Spencer-28] Spencer, Joel(9 August 2001).The Strange Logic of Random Graphs.Algorithms and Combinatorics. Vol. 22.Springer.pp. 3–4.ISBN 978-3-540-41654-8.

[Bollobas-29] Bollobás, Béla(8 October 2001).Random Graphs.Cambridge Studies in Advanced Mathematics. Vol. 73 (2nd ed.).Cambridge University Press.pp. 34–36.ISBN 978-0-521-79722-1.

[Gradel-30] Grädel, Eric; Kolaitis, Phokion G.;Libkin, Leonid;Marx, Maarten;Spencer, Joel;Vardi, Moshe Y.;Venema, Yde; Weinstein, Scott (11 June 2007).Finite Model Theory and Its Applications.Texts in Theoretical Computer Science (AnEATCSSeries).Springer.p. 298.ISBN 978-3-540-00428-8.

[Buckley-31] Buckley, Fred;Harary, Frank(21 January 1990).Distance in Graphs.Addison-Wesley.p. 109.ISBN 978-0-201-09591-3.

[Oxtoby-32] Oxtoby, John C. (1980).Measure and Category.Graduate Texts in Mathematics.Vol. 2 (2nd ed.). United States:Springer.pp. 59, 68.ISBN 978-0-387-90508-2.While Oxtoby does not explicitly define the term there,Babaihas borrowed it fromMeasure and Categoryin his chapter "Automorphism Groups, Isomorphism, Reconstruction" of Graham,GrötschelandLovász'sHandbook of Combinatorics(vol. 2), and Broer andTakensnote in their bookDynamical Systems and ChaosthatMeasure and Categorycompares 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).

[Baratchart-33] Baratchart, Laurent (1987). "Recent and New Results in Rational L²Approximation ". InCurtain, Ruth F.(ed.).Modelling, Robustness and Sensitivity Reduction in Control Systems.NATO ASI Series F. Vol. 34.Springer.p. 123.doi:10.1007/978-3-642-87516-8.ISBN 978-3-642-87516-8.

[Broer-34] Broer, Henk;Takens, Floris(28 October 2010).Dynamical Systems and Chaos.Applied Mathematical Sciences. Vol. 172.Springer.p. 245.doi:10.1007/978-1-4419-6870-8.ISBN 978-1-4419-6870-8.

[Sharkovsky-35] Sharkovsky, A. N.; Kolyada, S. F.; Sivak, A. G.; Fedorenko, V. V. (30 April 1997).Dynamics of One-Dimensional Maps.Mathematics and Its Applications. Vol. 407.Springer.p. 33.doi:10.1007/978-94-015-8897-3.ISBN 978-94-015-8897-3.

[Yuan-36] Yuan, George Xian-Zhi (9 February 1999).KKM Theory and Applications in Nonlinear Analysis.Pure and Applied Mathematics; A Series of Monographs and Textbooks.Marcel Dekker.p. 21.ISBN 978-0-8247-0031-7.

[Albertini-37] Albertini, Francesca;Sontag, Eduardo D.(1 September 1991). "Transitivity and Forward Accessibility of Discrete-Time Nonlinear Systems". In Bonnard, Bernard; Bride, Bernard; Gauthier, Jean-Paul; Kupka, Ivan (eds.).Analysis of Controlled Dynamical Systems.Progress in Systems and Control Theory. Vol. 8.Birkhäuser.p. 29.doi:10.1007/978-1-4612-3214-8.ISBN 978-1-4612-3214-8.

[Fuente-38] De la Fuente, Angel (28 January 2000).Mathematical Models and Methods for Economists.Cambridge University Press.p. 217.ISBN 978-0-521-58529-3.

[Komjath-40] Komjáth, Péter;Totik, Vilmos(2 May 2006).Problems and Theorems in Classical Set Theory.Problem Books in Mathematics. United States:Springer.p. 75.ISBN 978-0387-30293-5.

[Salzmann-41] 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. Vol. 112.Cambridge University Press.p.155.ISBN 978-0-521-86516-6.

[Schoutens-42] Schoutens, Hans (2 August 2010).The Use of Ultraproducts in Commutative Algebra.Lecture Notes in Mathematics.Vol. 1999.Springer.p. 8.doi:10.1007/978-3-642-13368-8.ISBN 978-3-642-13367-1.

[Rautenberg-43] Rautenberg, Wolfgang(17 December 2009).A Concise to Mathematical Logic.Universitext (3rd ed.).Springer.pp. 210–212.doi:10.1007/978-1-4419-1221-3.ISBN 978-1-4419-1221-3.

[Schwartzman-4] Schwartzman, Steven (1 May 1994).The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English.Spectrum Series.Mathematical Association of America.p.22.ISBN 978-0-88385-511-9.

[Clapham-9] Clapham, Christopher; Nicholson, James (7 June 2009).The Concise Oxford Dictionary of mathematics.Oxford Paperback References (4th ed.).Oxford University Press.p. 38.ISBN 978-0-19-923594-0.

[James-11] James, Robert C.(31 July 1992).Mathematics Dictionary(5th ed.).Chapman & Hall.p. 269.ISBN 978-0-412-99031-1.

[Bityutskov-15] Bityutskov, Vadim I. (30 November 1987)."Almost-everywhere".InHazewinkel, Michiel(ed.).Encyclopaedia of Mathematics.Vol. 1.Kluwer Academic Publishers.p. 153.doi:10.1007/978-94-015-1239-8.ISBN 978-94-015-1239-8.

[Ito2-16] Itô, Kiyosi,ed. (4 June 1993).Encyclopedic Dictionary of Mathematics.Vol. 2 (2nd ed.). Kingsport:MIT Press.p. 1267.ISBN 978-0-262-09026-1.

[RealTrans-19] "Almost All Real Numbers are Transcendental - ProofWiki".proofwiki.org.Retrieved2019-11-11.

[Weisstein-24] Weisstein, Eric W."Almost All".MathWorld.See alsoWeisstein, Eric W.(25 November 1988).CRC Concise Encyclopedia of Mathematics(1st ed.).CRC Press.p. 41.ISBN 978-0-8493-9640-3.

[Ito1-39] Itô, Kiyosi,ed. (4 June 1993).Encyclopedic Dictionary of Mathematics.Vol. 1 (2nd ed.). Kingsport:MIT Press.p. 67.ISBN 978-0-262-09026-1.

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Meaning in measure theory

Meaning in number theory

Meaning in graph theory

Meaning in topology

Meaning in algebra

Proofs

See also

References

Primary sources

Secondary sources