limit along a filter

Definition 1.

Let $\mathrm{F}$ be a filter on $\mathrm{N}$ and $\mathrm{(}{x}_n\mathrm{)}$ be a sequence in a metricspace $\mathrm{(}X\mathrm{,}d\mathrm{)}$. We say that $L$ is the $ℱ$-limit of $\mathrm{(}{x}_n\mathrm{)}$if

for every $\epsilon \mathrm{>}\mathrm{0}$.

The name along $\mathrm{F}$ is used as well.

In the usual definition of limit one requires all sets $A(\epsilon )$ tobe cofinite - i.e. they have to be large. In the definition of$ℱ$-limit we simply choose which sets are considered to be large- namely the sets from the filter $ℱ$.

Remarks

This notion shouldn’t be confused with the notion of limit of afilter (http://planetmath.org/filter) defined in general topology.

Let us note that the same notion is defined by some authors usingthe dual notion of ideal instead of filter and, of course, allresults can be reformulated using ideals as well. For thisapproach see e.g. [4].

Examples

Limit along the Fréchet filter, which consist of complements offinite sets, is the usual limit of a sequence.

Limit of the sequence $(x_n)$ along the principal filter${ℱ}_k=\{A\subseteq \mathbb{N};k\in A\}$ is $x_k$.

If we put $ℱ=\{\mathbb{N}\setminus A:d(A)=0\}$, where $d$ denotes theasymptotic density, then it can be shown that $ℱ$ is a filter. Inthis case $ℱ$-convergence is known as statistical convergence.