Set-theoretic limit


In mathematics, the limit of a sequence of sets A1, A2,... is a set whose elements are determined by the sequence in either of two equivalent ways: ' by upper and lower bounds on the sequence that converge monotonically to the same set and ' by convergence of a sequence of indicator functions which are themselves real-valued. As is the case with sequences of other objects, convergence is not necessary or even usual.
More generally, again analogous to real-valued sequences, the less restrictive limit infimum and limit supremum of a set sequence always exist and can be used to determine convergence: the limit exists if the limit infimum and limit supremum are identical.. Such set limits are essential in measure theory and probability.
It is a common misconception that the limits infimum and supremum described here involve sets of accumulation points, that is, sets of x = limk→∞xk, where each xk is in some Ank. This is only true if convergence is determined by the discrete metric. This article is restricted to that situation as it is the only one relevant for measure theory and probability. See the examples below.

Definitions

The two definitions

Suppose that is a sequence of sets. The two equivalent definitions are as follows.
To see the equivalence of the definitions, consider the limit infimum. The use of DeMorgan's rule below explains why this suffices for the limit supremum. Since indicator functions take only values 0 and 1, if and only if 1An takes value 0 only finitely many times. Equivalently, if and only if there exists n such that the element is in Am for every m ≥ n, which is to say if and only if x ∉ An for only finitely many n.
Therefore, x is in the iff x is in all except finitely many An. For this reason, a shorthand phrase for the limit infimum is "x ∈ An all except finitely often", typically expressed by "An a.e.f.o.".
Similarly, an element x is in the limit supremum if, no matter how large n is there exists m ≥ n such that the element is in Am. That is, x is in the limit supremum iff x is in infinitely many An. For this reason, a shorthand phrase for the limit supremum is "x ∈ An infinitely often", typically expressed by "An i.o.".
To put it another way, the limit infimum consists of elements that "eventually stay forever", while the limit supremum consists of elements that "never leave forever".

Monotone sequences

The sequence is said to be nonincreasing if each An+1 ⊂ An and nondecreasing if each An ⊂ An+1. In each of these cases the set limit exists. Consider, for example, a nonincreasing sequence. Then
From these it follows that
Similarly, if is nondecreasing then

Properties

Set limits, particularly the limit infimum and the limit supremum, are essential for probability and measure theory. Such limits are used to calculate the probabilities and measures of other, more purposeful, sets. For the following, is a probability space, which means is a σ-algebra of subsets of and is a probability measure defined on that σ-algebra. Sets in the σ-algebra are known as events.
If A1, A2,... is a monotone sequence of events in then exists and

Borel–Cantelli lemmas

In probability, the two Borel–Cantelli lemmas can be useful for showing that the limsup of a sequence of events has probability equal to 1 or to 0. The statement of the first Borel–Cantelli lemma is
The second Borel–Cantelli lemma is a partial converse:

Almost sure convergence

One of the most important applications to probability is for demonstrating the almost sure convergence of a sequence of random variables. The event that a sequence of random variables Y1, Y2,... converges to another random variable Y is formally expressed as. It would be a mistake, however, to write this simply as a limsup of events. That is, this is not the event ! Instead, the complement of the event is
Therefore,