In
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 (
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
) 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
where each
is in some
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,
if there is
such that
for all
). 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
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
and
If these two sets are equal, then the set-theoretic limit of the sequence
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
equal
if
and
otherwise. Define
and
where the expressions inside the brackets on the right are, respectively, the
limit infimum and
limit supremum of the real-valued sequence
Again, if these two sets are equal, then the set-theoretic limit of the sequence
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
and
if and only if
takes value
only finitely many times. Equivalently,
if and only if there exists
such that the element is in
for every
which is to say if and only if
for only finitely many
Therefore,
is in the
if and only if
is in all but finitely many
For this reason, a shorthand phrase for the limit infimum is "
is in
all but finitely often", typically expressed by writing "
a.b.f.o.".
Similarly, an element
is in the limit supremum if, no matter how large
is, there exists
such that the element is in
That is,
is in the limit supremum if and only if
is in infinitely many
For this reason, a shorthand phrase for the limit supremum is "
is in
infinitely often", typically expressed by writing "
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:
:
Monotone sequences
The sequence
is said to be nonincreasing if
for each
and nondecreasing if
for each
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
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
as
goes to infinity, exists for all
then
Otherwise, the limit for
does not exist.
* It can be shown that the limit infimum is contained in the limit supremum:
for example, simply by observing that
all but finitely often implies
infinitely often.
* Using the
monotonicity of
and of
* 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 ...
That is,
all but finitely often is the same as
finitely often.
* From the second definition above and the definitions for limit infimum and limit supremum of a real-valued sequence,
and
* Suppose
is a
-algebra of subsets of
That is,
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
then both
and
are elements of
Examples
* Let
Then
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