limit superior of sets

Let A1,A2, be a sequence of sets. The limit superior of sets is defined by

lim supAn=n=1k=nAk.

It is easy to see that xlim supAn if and only if xAn for infinitely many values of n. Because of this, in probability theory the notation [Ani.o.] is often used to refer to lim supAn, where i.o. stands for infinitely often.

The limit inferior of sets is defined by

lim infAn=n=1k=nAk,

and it can be shown that xlim infAn if and only if x belongs to An for all but finitely many values of n.

Title limit superior of sets
Canonical name LimitSuperiorOfSets
Date of creation 2013-03-22 13:13:22
Last modified on 2013-03-22 13:13:22
Owner Koro (127)
Last modified by Koro (127)
Numerical id 8
Author Koro (127)
Entry type Definition
Classification msc 28A05
Classification msc 60A99
Defines limit inferior of sets
Defines infinitely often
Defines i.o.