In mathematics, the limit of a sequence of sets (subsets of a common set X) is a set whose elements are determined by the sequence in either of two equivalent ways: (1) by upper and lower bounds on the sequence that converge monotonically to the same set (analogous to convergence of real-valued sequences) and (2) 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. (See below). 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 where each x_k is in some A_{n_k}. This is only true if convergence is determined by the discrete metric (that is, x_n \to x if there is N such that x_n = x for all n \geq N). This article is restricted to that situation as it is the only one relevant for measure theory and probability. See the examples below. (On the other hand, there are more general topological notions of set convergence that do involve accumulation points under different metrics or topologies.)

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 De Morgan's law below explains why this suffices for the limit supremum. Since indicator functions take only values 0 and 1, if and only if takes value 0 only finitely many times. Equivalently, if and only if there exists n such that the element is in A_m for every m \geq n, which is to say if and only if for only finitely many n. Therefore, x is in the if and only if x is in all but finitely many A_n. For this reason, a shorthand phrase for the limit infimum is "x is in A_n all but finitely often", typically expressed by writing "A_n a.b.f.o.". Similarly, an element x is in the limit supremum if, no matter how large n is, there exists m \geq n such that the element is in A_m. That is, x is in the limit supremum if and only if x is in infinitely many A_n. For this reason, a shorthand phrase for the limit supremum is "x is in A_n infinitely often", typically expressed by writing "A_n i.o.". To put it another way, the limit infimum consists of elements that "eventually stay forever" (are in set after n), while the limit supremum consists of elements that "never leave forever" (are in set after n). Or more formally:

Monotone sequences

The sequence is said to be nonincreasing if for each n, and nondecreasing if for each n. 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 The Cantor set is defined this way.

Properties

Examples

and so exists. and so does not exist, despite the fact that the left and right endpoints of the intervals converge to 0 and 1, respectively. is the set of all rational numbers between 0 and 1 (inclusive), since even for j < n and is an element of the above. Therefore, On the other hand, which implies In this case, the sequence does not have a limit. Note that is not the set of accumulation points, which would be the entire interval [0, 1] (according to the usual Euclidean metric).

Probability uses

Set limits, particularly the limit infimum and the limit supremum, are essential for probability and measure theory. Such limits are used to calculate (or prove) the probabilities and measures of other, more purposeful, sets. For the following, is a probability space, which means \mathcal{F} is a σ-algebra of subsets of X and \mathbb{P} is a probability measure defined on that σ-algebra. Sets in the σ-algebra are known as events. If is a monotone sequence of events in \mathcal{F} 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 (original) 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 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 the event ! Instead, the of the event is Therefore,