# Almost all

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In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null.

In contrast, "almost no" means "a negligible amount"; that is, "almost no elements of X" means "the elements of some negligible subset of X".

## Meanings in areas of mathematics

### Prevalent meaning

Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) but finitely many"; this use occurs in philosophy as well. Similarly, "almost all" can mean "all (elements of an uncountable set) but countably many".[sec 1]

Examples:

### Meaning in measure theory

When speaking about the reals, sometimes "almost all" means "all reals but a null set".[sec 2] Similarly, if S is some set of reals, "almost all numbers in S" can mean "all numbers in S but those in a null set". The real line can be thought of as a one-dimensional Euclidean space. In the more general case of an n-dimensional space (where n is a positive integer), these definitions can be generalised to "all points but those in a null set"[sec 3] or "all points in S but those in a null set" (this time, S is a set of points in the space). Even more generally, "almost all" is sometimes used in the sense of "almost everywhere" in measure theory,[sec 4] or in the closely related sense of "almost surely" in probability theory.[sec 5]

Examples:

### Meaning in number theory

In number theory, "almost all positive integers" can mean "the positive integers in a set whose natural density is 1"; that is, if A is a set of positive integers, and if the proportion of positive integers below n that are in A (out of all positive integers below n) tends to 1 as n tends to infinity, then almost all positive integers are in A.[sec 6] More generally, let S be an infinite set of positive integers, such as the set of even positive numbers or of primes. If A is a subset of S, and if the proportion of elements of S below n that are in A (out of all elements of S below n) tends to 1 as n tends to infinity, then it can be said that almost all elements of S are in A.

Examples:

• The natural density of cofinite sets of positive integers is 1, so each of them contains almost all positive integers.
• Almost all positive integers are composite.[sec 6][proof 1]
• Almost all even positive numbers can be expressed as the sum of two primes.:489
• Almost all primes are isolated. Moreover, for every positive integer g, almost all primes have prime gaps of more than g both to their left and to their right (no other primes between p−g and p+g).

### Meaning in graph theory

In graph theory, if A is a set of (finite labelled) graphs, it can be said to contain almost all graphs if the proportion of graphs with n vertices that are in A tends to 1 as n tends to infinity. However, it is sometimes easier to work with probabilities, so the definition is reformulated as follows; the proportion of graphs with n vertices that are in A equals the probability that a random graph with n vertices (chosen with the uniform distribution) is in A, 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. Therefore, equivalently to the preceding definition, A contains almost all graphs if the probability that a coin flip-generated graph with n vertices is in A tends to 1 as n tends to infinity. Sometimes the latter definition is modified so that the graph is chosen randomly in some other way, where not all graphs with n vertices have the same probability, 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.

Example:

### Meaning in topology

In topology and especially dynamical systems theory (including applications in economics), "almost all" of a topological space's points can mean "all of the space's points but those in a meagre set"; some use a more limited definition, where a subset only contains almost all of the space's points if it contains some open dense set.

Example:

### Meaning in algebra

In abstract algebra and mathematical logic, if U is an ultrafilter on a set X, "almost all elements of X" sometimes means "the elements of some element of U". For any partition of X into two disjoint sets, one of them necessarily contains almost all elements of X. It is possible to think of the elements of a filter on X as containing almost all elements of X even if it isn't an ultrafilter.

## Proofs

1. ^ According to the prime number theorem, the number of primes less than or equal to n is asymptotically equal to n/ln(n). Therefore, the proportion of primes is roughly ln(n)/n, which tends to 0 as n tends to infinity, so the proportion of composite numbers less than or equal to n tends to 1 as n tends to infinity.