Almost all

This article covers almost all of the term as meaning all but a finite number . For use in measurement and integration theory and in probability theory see almost everywhere or almost certainly .

In mathematics, almost all is usually an abbreviation for all but a finite number , mostly in connection with countable basic sets. It is said that a property is fulfilled by almost all elements of an infinite set , if only finitely many elements are not fulfilled. Subsets that contain almost all elements of a set are also called kofinite or kofinite , because their complement is finite. ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {E}}}$

But there are also different uses of the term.

Specialization in consequences

The property applies to almost all members of a sequence , if at most finitely many members are counterexamples. ${\displaystyle {\mathcal {E}}}$

This can also be characterized as follows: There is a sequence term from which the property applies to all subsequent terms.

Formal: . ${\displaystyle \exists N\in \mathbb {N} \colon \forall n>N\colon {\mathcal {E}}(a_{n})}$

This should not be confused with the demand , which means that for an infinite number of true followers. This is a really weaker requirement, because it does not exclude the fact that it does not hold for an infinite number of sequence members either . ${\displaystyle \forall N\in \mathbb {N} \colon \exists n>N\colon {\mathcal {E}}(a_{n})}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {E}}}$

Examples and counterexamples

1. There are infinitely many natural numbers divisible by 3 , because for every given natural number one can find a larger one that is divisible by 3, because one of the numbers or must have this property. But there are also infinitely many numbers that cannot be divided by 3. The term almost all does not apply here. ${\displaystyle N}$ ${\displaystyle N+1,N+2}$ ${\displaystyle N+3}$
2. Almost all of the positive numbers divisible by three are greater than 15 trillion, because there are finitely many (namely 5 trillion) numbers divisible by 3 that are not greater than 15 trillion; the infinitely many other numbers divisible by three are greater than 15 trillion.
3. A real sequence of numbers : ${\displaystyle (a_{n}\mid n\in \mathbb {N} )}$
   • has an accumulation point if for each there are infinitely many terms in the open interval . However, there can also be an infinite number of sequence elements outside the interval and there can even be further accumulation points. ${\displaystyle b}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle ]b-\varepsilon ,b+\varepsilon [}$
   • has the limit value if for each almost all elements of the sequence are in the open interval - i.e. only finitely many outside. ${\displaystyle a}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle ]b-\varepsilon ,b+\varepsilon [}$
4. There are significantly more real numbers than whole numbers. Nevertheless, one cannot say that almost all real numbers are not whole, since there are an infinite number of whole real numbers (even if only countably many).
5. If there is any index set and if one has a (e.g. real) number for each index , then one can only meaningfully define the sum of all without resorting to a concept of convergence if almost all are, namely as the sum of the finitely many numbers different from 0 . ${\displaystyle I}$ ${\displaystyle i\in I}$ ${\displaystyle a_{i}}$ ${\displaystyle a_{i}}$ ${\displaystyle a_{i}=0}$

generalization

Be a set filter on a set . A property applies -almost all over (or for -almost all in ) when the amount of those in which the property that meet the filter is located. ${\displaystyle F}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle F}$ ${\displaystyle X}$ ${\displaystyle F}$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle F}$

The term almost all explained above is exactly the term - almost all for the special case of the Fréchet filter consisting of all cofinite subsets of . ${\displaystyle F}$ ${\displaystyle X}$

In measure theory , another special case is often considered; If the filter is based on those sets whose complement has measure 0, then -almost all means the same thing as almost everywhere and is useful because almost everywhere you can not refer to elements of sets. ${\displaystyle F}$ ${\displaystyle F}$

Number theory

In number theory one also speaks of the fact that almost all natural numbers are in a set , if , where the number of elements is in with . This can also be expressed with the Landau symbols :, with . In addition to the natural numbers, other infinite sets can also be chosen as a basis. For example, almost all natural numbers are composite and almost all prime numbers are isolated . ${\displaystyle A}$ ${\displaystyle \lim _{n\to \infty }{\frac {A(n)}{n}}=1}$ ${\displaystyle A(n)}$ ${\displaystyle x}$ ${\displaystyle A}$ ${\displaystyle x<n}$ ${\displaystyle {\bar {A}}(n)\in o(n)}$ ${\displaystyle {\bar {A}}(n)=n-A(n)}$

Individual evidence

1. ↑ Hardy, Wright, An Introduction to the theory of numbers, 4th edition, Oxford, Clarendon Press 1975, p. 8