Almost all

Almost all is in mathematics shorthand for all but finitely many, mostly in connection with countable basic quantities. It is said that a property may be fulfilled by almost all the elements of an infinite set, if only finitely many elements do not meet. Subsets which contain almost all elements of a set are also called koendlich or kofinit because its complement is finite.

Specialization on Follow

The feature applies to almost all the members of a sequence when only finitely many terms of the sequence are counter-examples.

This can also be characterized as follows: There are a follower, is true of the of the property for all subsequent links. Formal:.

This should not be confused with the request, which means that applies to an infinite number of followers. This is a " really weak " requirement, because they do not exclude the possibility that for infinitely many terms of the sequence also does not apply.

Examples and counter-examples

• Has an accumulation point b if for every ε > 0 infinitely many terms of the sequence lie in the interval (b- ε, b ε ). However, it is also infinitely many terms of the sequence lie outside this interval; it may even be other points of accumulation.
• Has the limit a, if for every ε > 0, almost all followers in the interval (a- ε, a ε ) lie - so only finitely many outside.

Generalization

Let F be a lot of filters on a set X. A property applies F -almost everywhere on X (or for F- almost all x in X ) if the quantity is that x in X that satisfy the property in the filter F.

The above- stated term, almost all is exactly the term F- almost all existing for the special case of kofiniten of all subsets of X Fréchet filter.

In measure theory, another special case is often seen; when the filter F of those amounts based sets whose complement has measure 0, this means F- almost all the same as almost everywhere.

• Mathematical concept