In mathematics, the limit of a sequence of sets (subsets) 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.

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. (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

is a sequence of sets. The two equivalent definitions are as follows.

• Using union and intersection: define exists and is equal to that common set. Either set as described above can be used to get the limit, and there may be other means to get the limit as well.

• Using indicator functions otherwise. Define where the expressions inside the brackets on the right are, respectively, the limit infimum and limit supremum exists and is equal to that common set, and either set as described above can be used to get the limit.

To see the equivalence of the definitions, consider the limit infimum. The use of De Morgan's law a.b.f.o.".

i.o.".

To put it another way, the limit infimum consists of elements that "eventually stay forever" (are in set after ), while the limit supremum consists of elements that "never leave forever" (are in set after ). Or more formally:

1. {|

. |}

Monotone sequences

is said to be nonincreasing and nondecreasing

The Cantor set is defined this way.

Properties

• does not exist.

• infinitely often.

• Using the monotonicity

• By using De Morgan's law twice, with set complement finitely often.

• is a -algebra is nonempty and is closed under complement and under unions and intersections of countably many

Examples

• Then

exists.

• Then

does not exist, despite the fact that the left and right endpoints of the intervals converge to 0 and 1, respectively.

• Then

is the set of all rational numbers (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 is a probability space is a σ-algebra is a probability measure defined on that σ-algebra. Sets in the σ-algebra are known as events.

is a monotone sequence

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 It would be a mistake, however, to write this simply as a limsup of events. That is, this ! Instead, the

See also

References