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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the limit of a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of sets A_1, A_2, \ldots (
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s 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 function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
s 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 In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
and
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
. It is a common misconception that the limits infimum and supremum described here involve sets of accumulation points, that is, sets of x = \lim_ x_k, where each x_k is in some A_. This is only true if convergence is determined by the
discrete metric Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
(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 Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...
or topologies.)


Definitions


The two definitions

Suppose that \left(A_n\right)_^\infty is a sequence of sets. The two equivalent definitions are as follows. * Using union and
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
: define \liminf_ A_n = \bigcup_ \bigcap_ A_j and \limsup_ A_n = \bigcap_ \bigcup_ A_j If these two sets are equal, then the set-theoretic limit of the sequence A_n 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 function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
s: let \mathbb_(x) equal 1 if x \in A_n, and 0 otherwise. Define \liminf_ A_n = \Bigl\ and \limsup_ A_n = \Bigl\, where the expressions inside the brackets on the right are, respectively, the limit infimum and limit supremum of the real-valued sequence \mathbb_(x). Again, if these two sets are equal, then the set-theoretic limit of the sequence A_n 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 In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathemat ...
below explains why this suffices for the limit supremum. Since indicator functions take only values 0 and 1, \liminf_ \mathbb_(x) = 1 if and only if \mathbb_(x) takes value 0 only finitely many times. Equivalently, x \in \bigcup_ \bigcap_ A_j 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 x \not\in A_n for only finitely many n. Therefore, x is in the \liminf_ A_n 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 \left(A_n\right) is said to be nonincreasing if A_ \subseteq A_n for each n, and nondecreasing if A_n \subseteq A_ for each n. In each of these cases the set limit exists. Consider, for example, a nonincreasing sequence \left(A_n\right). Then \bigcap_ A_j = \bigcap_ A_j \text \bigcup_ A_j = A_n. From these it follows that \liminf_ A_n = \bigcup_ \bigcap_ A_j = \bigcap_ A_j = \bigcap_ \bigcup_ A_j = \limsup_ A_n. Similarly, if \left(A_n\right) is nondecreasing then \lim_ A_n = \bigcup_ A_j. The
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Throu ...
is defined this way.


Properties

* If the limit of \mathbb_(x), as n goes to infinity, exists for all x then \lim_ A_n = \left\. Otherwise, the limit for \left(A_n\right) does not exist. * It can be shown that the limit infimum is contained in the limit supremum: \liminf_ A_n \subseteq \limsup_ A_n, for example, simply by observing that x \in A_n all but finitely often implies x \in A_n infinitely often. * Using the monotonicity of B_n = \bigcap_ A_j and of C_n = \bigcup_ A_j, \liminf_ A_n = \lim_\bigcap_ A_j \quad \text \quad \limsup_ A_n = \lim_ \bigcup_ A_j. * By using
De Morgan's law In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathemat ...
twice, with
set complement In set theory, the complement of a set , often denoted by A^c (or ), is the set of elements not in . When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set , the absolute complemen ...
A^c := X \setminus A, \liminf_ A_n = \bigcup_n \left(\bigcup_ A_j^c\right)^c = \left(\bigcap_n \bigcup_ A_j^c\right)^c = \left(\limsup_ A_n^c\right)^c. That is, x \in A_n all but finitely often is the same as x \not\in A_n finitely often. * From the second definition above and the definitions for limit infimum and limit supremum of a real-valued sequence, \mathbb_(x) = \liminf_\mathbb_(x) = \sup_ \inf_ \mathbb_(x) and \mathbb_(x) = \limsup_ \mathbb_(x) = \inf_ \sup_ \mathbb_(x). * Suppose \mathcal is a -algebra of subsets of X. That is, \mathcal is
nonempty In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, whi ...
and is closed under complement and under unions and intersections of countably many sets. Then, by the first definition above, if each A_n \in \mathcal then both \liminf_ A_n and \limsup_ A_n are elements of \mathcal.


Examples

* Let A_n = \left(- \tfrac, 1 - \tfrac\right]. Then \liminf_ A_n = \bigcup_n \bigcap_ \left(-\tfrac, 1 - \tfrac \right] = \bigcup_n \left , 1 - \tfrac\right= , 1) and \limsup_ A_n = \bigcap_n \bigcup_\left(-\tfrac, 1 - \tfrac\right= \bigcap_n \left(- \tfrac, 1\right) = , 1) so \lim_ A_n = [0, 1) exists. * Change the previous example to A_n = \left(\tfrac, 1 - \tfrac\right Then \liminf_ A_n = \bigcup_n \bigcap_ \left(\tfrac, 1-\tfrac\right] = \bigcup_n \left(\tfrac, 1 - \tfrac\right] = (0, 1) and \limsup_ A_n = \bigcap_n \bigcup_ \left(\tfrac, 1 - \tfrac\right] = \bigcap_n \left(-\tfrac, 1 + \tfrac\right] = [0, 1], so \lim_ A_n does not exist, despite the fact that the left and right endpoints of the Interval (mathematics), intervals converge to 0 and 1, respectively. * Let A_n = \left\. Then \bigcup_ A_j = \Q\cap ,1/math> is the set of all
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s between 0 and 1 (inclusive), since even for j < n and 0 \leq k \leq j, \tfrac = \tfrac is an element of the above. Therefore, \limsup_ A_n = \Q \cap
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
On the other hand, \bigcap_ A_j = \, which implies \liminf_ A_n = \. In this case, the sequence A_1, A_2, \ldots does not have a limit. Note that \lim_ A_n is not the set of accumulation points, which would be the entire interval
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> (according to the usual
Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is oc ...
).


Probability uses

Set limits, particularly the limit infimum and the limit supremum, are essential for
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
and
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
. Such limits are used to calculate (or prove) the probabilities and measures of other, more purposeful, sets. For the following, (X,\mathcal,\mathbb) is a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models ...
, which means \mathcal is a
σ-algebra In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with a ...
of subsets of X and \mathbb is a
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
defined on that σ-algebra. Sets in the σ-algebra are known as events. If A_1, A_2, \ldots is a monotone sequence of events in \mathcal then \lim_ A_n exists and \mathbb\left(\lim_ A_n\right) = \lim_ \mathbb\left(A_n\right).


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 Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
is for demonstrating the
almost sure convergence In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
of a sequence of
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s. The event that a sequence of random variables Y_1, Y_2, \ldots converges to another random variable Y is formally expressed as \left\. It would be a mistake, however, to write this simply as a limsup of events. That is, this the event \limsup_ \left\! Instead, the of the event is \begin \left\ &= \left\\\ &= \bigcup_ \bigcap_ \bigcup_ \left\ \\ &= \lim_ \limsup_ \left\. \end Therefore, \mathbb\left(\left\\right) = \lim_ \mathbb\left(\limsup_ \left\\right).


See also

* *


References

{{reflist, group=note Set theory Probability theory