Editing Almost all (section)

==Meanings in different areas of mathematics==
===Prevalent meaning===
{{further|Cofinite set}}
Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an [[infinite set]]) except for [[finite set|finite]]ly many".{{r|Cahen1|Cahen2}} This use occurs in philosophy as well.{{r|Gardenfors}} Similarly, "almost all" can mean "all (elements of an [[uncountable set]]) except for [[countable set|countably]] many".{{r|Schwartzman|group=sec}}

Examples:
* Almost all positive integers are greater than 10<sup>12</sup>.{{r|Courant|page=293}}
* Almost all [[prime number]]s are odd (2 is the only exception).<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>
* Almost all [[polyhedra]] are [[regular polyhedron#The regular polyhedra|irregular]] (as there are only nine exceptions: the five [[platonic solid]]s and the four [[Kepler–Poinsot polyhedron|Kepler–Poinsot polyhedra]]).
* If <var>P</var> is a nonzero [[polynomial]], then <var>P(x)</var> &ne; 0 for almost all <var>x</var> (if not all ''x'').

===Meaning in measure theory===
{{further|Almost everywhere}}
[[File:CantorEscalier.svg|thumb|right|250px| The [[Cantor function]] as a function that has zero derivative almost everywhere]]

When speaking about the [[real number|reals]], sometimes "almost all" can mean "all reals except for a [[null set]]".{{r|Korevaar|Natanson}}{{r|Clapham|group=sec}} Similarly, if <var>S</var> is some set of reals, "almost all numbers in <var>S</var>" can mean "all numbers in <var>S</var> except for those in a null set".{{r|Sohrab}} The [[real line]] can be thought of as a one-dimensional [[Euclidean space]]. In the more general case of an <var>n</var>-dimensional space (where <var>n</var> is a positive integer), these definitions can be [[generalised]] to "all points except for those in a null set"{{r|James|group=sec}} or "all points in <var>S</var> except for those in a null set" (this time, <var>S</var> is a set of points in the space).{{r|Helmberg}} Even more generally, "almost all" is sometimes used in the sense of "[[almost everywhere]]" in [[measure theory]],{{r|Vestrup|Billingsley}}{{r|Bityutskov|group=sec}} or in the closely related sense of "[[almost surely]]" in [[probability theory]].{{r|Billingsley}}{{r|Ito2|group=sec}}

Examples:
* In a [[measure space]], such as the real line, countable sets are null. The set of [[rational number]]s is countable, so almost all real numbers are irrational.{{r|Niven}}
* Georg [[Cantor's first set theory article]] proved that the set of [[algebraic number]]s is countable as well, so almost all reals are [[transcendental number|transcendental]].{{r|Baker}}{{r|group=sec|RealTrans}}
* Almost all reals are [[normal number|normal]].{{r|Granville}}
* The [[Cantor set]] is also null. Thus, almost all reals are not in it even though it is uncountable.{{r|Korevaar}}
* The derivative of the [[Cantor function]] is 0 for almost all numbers in the [[unit interval]].{{r|Burk}} It follows from the previous example because the Cantor function is [[locally constant function|locally constant]], and thus has derivative 0 outside the Cantor set.

===Meaning in number theory===
{{further|Asymptotically almost surely}}
In [[number theory]], "almost all positive integers" can mean "the positive integers in a set whose [[natural density]] is 1". That is, if <var>A</var> is a set of positive integers, and if the proportion of positive integers in ''A'' below <var>n</var> (out of all positive integers below <var>n</var>) [[limit of a sequence|tends to]] 1 as <var>n</var> tends to infinity, then almost all positive integers are in <var>A</var>.{{r|Hardy1|Hardy2}}{{r|Weisstein|group=sec}} 

More generally, let <var>S</var> be an infinite set of positive integers, such as the set of even positive numbers or the set of [[prime number|primes]], if <var>A</var> is a subset of <var>S</var>, and if the proportion of elements of <var>S</var> below <var>n</var> that are in <var>A</var> (out of all elements of <var>S</var> below <var>n</var>) tends to 1 as <var>n</var> tends to infinity, then it can be said that almost all elements of <var>S</var> are in <var>A</var>.

Examples:
* The natural density of [[cofinite set]]s of positive integers is 1, so each of them contains almost all positive integers.
* Almost all positive integers are [[composite number|composite]].{{r|Weisstein|group=sec}}{{refn |group=proof |The [[prime number theorem]] shows that the number of primes less than or equal to <var>n</var> is asymptotically equal to <var>n</var>/ln(<var>n</var>). Therefore, the proportion of primes is roughly ln(<var>n</var>)/<var>n</var>, which tends to 0 as <var>n</var> tends to [[infinity]], so the proportion of composite numbers less than or equal to <var>n</var> tends to 1 as <var>n</var> tends to infinity.{{r|Hardy2}}}}
* Almost all even positive numbers can be expressed as the sum of two primes.{{r|Courant|page=489}}
* Almost all primes are [[twin prime#Isolated prime|isolated]]. Moreover, for every positive integer {{mvar|g}}, almost all primes have [[prime gap]]s of more than {{mvar|g}} both to their left and to their right; that is, there is no other prime between {{math|''p'' − ''g''}} and {{math|''p'' + ''g''}}.{{r|Prachar}}

===Meaning in graph theory===
In [[graph theory]], if <var>A</var> is a set of (finite [[graph labeling|labelled]]) [[graph (discrete mathematics)|graph]]s, it can be said to contain almost all graphs, if the proportion of graphs with <var>n</var> vertices that are in <var>A</var> tends to 1 as <var>n</var> tends to infinity.{{r|Babai}} However, it is sometimes easier to work with probabilities,{{r|Spencer}} so the definition is reformulated as follows. The proportion of graphs with <var>n</var> vertices that are in <var>A</var> equals the probability that a random graph with <var>n</var> vertices (chosen with the [[discrete uniform distribution|uniform distribution]]) is in <var>A</var>, 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.{{r|Bollobas}} Therefore, equivalently to the preceding definition, the set <var>''A''</var> contains almost all graphs if the probability that a coin-flip–generated graph with <var>n</var> vertices is in <var>A</var> tends to 1 as <var>n</var> tends to infinity.{{r|Spencer|Gradel}} Sometimes, the latter definition is modified so that the graph is chosen randomly in some [[random graph#Models|other way]], where not all graphs with <var>n</var> vertices have the same probability,{{r|Bollobas}} 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.{{r|Spencer}}

Example:
* Almost all graphs are [[asymmetric graph|asymmetric]].{{r|Babai}}
* Almost all graphs have [[diameter (graph theory)|diameter]] 2.{{r|Buckley}}

===Meaning in topology===
In [[topology]]{{r|Oxtoby}} and especially [[dynamical systems theory]]{{r|Baratchart|Broer|Sharkovsky}} (including applications in economics),{{r|Yuan}} "almost all" of a [[topological space]]'s points can mean "all of the space's points except for those in a [[meagre set]]". Some use a more limited definition, where a subset contains almost all of the space's points only if it contains some [[Open set|open]] [[dense set]].{{r|Broer|Albertini|Fuente}}

Example:
* Given an [[hyperconnected space|irreducible]] [[algebraic variety]], the [[Property (mathematics)|properties]] that hold for almost all points in the variety are exactly the [[generic property|generic properties]].{{r|Ito1|group=sec}} This is due to the fact that in an irreducible algebraic variety equipped with the [[Zariski topology]], all nonempty open sets are dense.

===Meaning in algebra===
In [[abstract algebra]] and [[mathematical logic]], if <var>U</var> is an [[Ultrafilter#Special case: ultrafilter on the powerset of a set|ultrafilter]] on a set <var>X,</var> "almost all elements of <var>X</var>" sometimes means "the elements of some ''element'' of <var>U</var>".{{r|Komjath|Salzmann|Schoutens|Rautenberg}} For any [[Partition of a set|partition]] of <var>X</var> into two [[disjoint sets]], one of them will necessarily contain almost all elements of <var>X.</var> It is possible to think of the elements of a [[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}}