limit along a filter

Definition 1.

Let F be a filter on N and (xn) be a sequencePlanetmathPlanetmath in a metric space (X,d). We say that L is the -limit of (xn) if


for every ε>0.

The name along F is used as well.

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


This notion shouldn’t be confused with the notion of limit of a filter ( defined in general topology.

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


Limit along the Fréchet filter, which consist of complementsPlanetmathPlanetmath of finite setsMathworldPlanetmath, is the usual limit of a sequence.

Limit of the sequence (xn) along the principal filterPlanetmathPlanetmathPlanetmath k={A;kA} is xk.

If we put ={A:d(A)=0}, where d denotes the asymptotic density, then it can be shown that is a filter. In this case -convergence is known as statistical convergence.


  • 1 M. A. Alekseev, L. Yu. Glebsky, and E. I. Gordon, On approximations of groups, group actions and Hopf algebrasPlanetmathPlanetmathPlanetmath, Journal of Mathematical Sciences 107 (2001), no. 5, 4305–4332.
  • 2 B. Balcar and P. Štěpánek, Teorie množin, Academia, Praha, 1986 (Czech).
  • 3 K. Hrbacek and T. Jech, Introduction to set theoryMathworldPlanetmath, Marcel Dekker, New York, 1999.
  • 4 P. Kostyrko, T. Šalát, and W. Wilczyński, I-convergence, Real Anal. Exchange 26 (2000-2001), 669–686.
Title limit along a filter
Canonical name LimitAlongAFilter
Date of creation 2013-03-22 15:32:20
Last modified on 2013-03-22 15:32:20
Owner kompik (10588)
Last modified by kompik (10588)
Numerical id 6
Author kompik (10588)
Entry type Definition
Classification msc 03E99
Classification msc 40A05
Synonym limit along filter
Synonym F-limit
Related topic Filter
Defines limit along a filter