In mathematics, the limit of a sequence of sets
) 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.
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
This is only true if convergence is determined by the discrete metric (that is,
This article is restricted to that situation as it is the only one relevant for measure theory and probability.
(On the other hand, there are more general topological notions of set convergence that do involve accumulation points under different metrics or topologies.)
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
For this reason, a shorthand phrase for the limit infimum is "
all but finitely often", typically expressed by writing "
is in the limit supremum if, no matter how large
For this reason, a shorthand phrase for the limit supremum is "
infinitely often", typically expressed by writing "
To put it another way, the limit infimum consists of elements that "eventually stay forever" (are in each set after some
), while the limit supremum consists of elements that "never leave forever" (are in some set after each
In each of these cases the set limit exists.
The Cantor set is defined this way.
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
is not the set of accumulation points, which would be the entire interval
Such limits are used to calculate (or prove) the probabilities and measures of other, more purposeful, sets.
is a probability measure defined on that σ-algebra.
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.
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
It would be a mistake, however, to write this simply as a limsup of events.