Set-theoretic limit explained

A _{1,} A _{2,} \ldots (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

x = \lim _{k} x _{k,} 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

\left(A _{n\right)} ^{inf ty} ^{n =1} is a sequence of sets. The two equivalent definitions are as follows.

• Using union and intersection: define^{[1] [2]} $$\backslash liminf\_\; A\_n\; =\; \backslash bigcup\_\; \backslash bigcap\_\; A\_j$$ and $$\backslash limsup\_\; A\_n\; =\; \backslash bigcap\_\; \backslash bigcup\_\; A\_j$$ 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 functions: let

1 _{A _{n}} (x) equal 1 if x \in A _{n,} and 0 otherwise. Define^[1] $$\backslash liminf\_\; A\_n\; =\; \backslash Bigl\backslash$$ and $$\backslash limsup\_\; A\_n\; =\; \backslash Bigl\backslash ,$$ where the expressions inside the brackets on the right are, respectively, the limit infimum and limit supremum of the real-valued sequence 1 _{A _{n}} (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 below explains why this suffices for the limit supremum. Since indicator functions take only values

0 and 1 , \liminf _{n} 1 _{A _{n}} (x) = 1 if and only if 1 _{A _{n}} (x) takes value 0 only finitely many times. Equivalently, $x\; \backslash in\; \backslash bigcup\_\; \backslash bigcap\_\; A\_j$ 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 _{n} 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:

$\backslash lim\_A\_n\; =\; L\; \backslash quad\; \backslash Longleftrightarrow$	for every x\in L       there is a n _{0\in\N} with x\in A _{n} for all n\ge n _{0} and
	for every y\in X\setminus\	L there is a p _{0\in\N} with y\not\in A _{p} for all p\ge p _{0} .

Monotone sequences

The sequence

\left(A _{n\right)} is said to be nonincreasing if A _{n+1} \subseteq A _{n} for each n, and nondecreasing if A _{n} \subseteq A _{n+1} for each n. In each of these cases the set limit exists. Consider, for example, a nonincreasing sequence \left(A _{n\right).} Then $$\backslash bigcap\_\; A\_j\; =\; \backslash bigcap\_\; A\_j\; \backslash text\; \backslash bigcup\_\; A\_j\; =\; A\_n.$$From these it follows that$$\backslash liminf\_\; A\_n\; =\; \backslash bigcup\_\; \backslash bigcap\_\; A\_j\; =\; \backslash bigcap\_\; A\_j\; =\; \backslash bigcap\_\; \backslash bigcup\_\; A\_j\; =\; \backslash limsup\_\; A\_n.$$Similarly, if \left(A _{n\right)} is nondecreasing then $$\backslash lim\_\; A\_n\; =\; \backslash bigcup\_\; A\_j.$$

The Cantor set is defined this way.

Properties

• If the limit of

1 _{A _{n}} (x), as n goes to infinity, exists for all x then $$\backslash lim\_\; A\_n\; =\; \backslash left\backslash .$$ 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: $$\backslash liminf\_\; A\_n\; \backslash subseteq\; \backslash limsup\_\; A\_n,$$ 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 $B\_n\; =\; \backslash bigcap\_\; A\_j$ and of $C\_n\; =\; \backslash bigcup\_\; A\_j,$ $$\backslash liminf\_\; A\_n\; =\; \backslash lim\_\backslash bigcap\_\; A\_j\; \backslash quad\; \backslash text\; \backslash quad\; \backslash limsup\_\; A\_n\; =\; \backslash lim\_\; \backslash bigcup\_\; A\_j.$$

A ^{c} : = X \setminus A, $$\backslash liminf\_\; A\_n\; =\; \backslash bigcup\_n\; \backslash left(\backslash bigcup\_\; A\_j^c\backslash right)^c=\; \backslash left(\backslash bigcap\_n\; \backslash bigcup\_\; A\_j^c\backslash right)^c=\; \backslash left(\backslash limsup\_\; A\_n^c\backslash right)^c.$$ 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, $$\backslash mathbb\_(x)\; =\; \backslash liminf\_\backslash mathbb\_(x)\; =\; \backslash sup\_\; \backslash inf\_\; \backslash mathbb\_(x)$$ and $$\backslash mathbb\_(x)\; =\; \backslash limsup\_\; \backslash mathbb\_(x)\; =\; \backslash inf\_\; \backslash sup\_\; \backslash mathbb\_(x).$$
• Suppose

l{F} is a -algebra of subsets of X. That is, l{F} is nonempty 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 l{F} then both \liminf _{n} A _{n} and \limsup _{n} A _{n} are elements of l{F}.

Examples

• Let

A _{n} = \left( - \tfrac{ 1 }{n}, 1 - \tfrac{ 1 }{n}\right]. Then Then$$\backslash liminf\_\; A\_n\; =\; \backslash bigcup\_n\; \backslash bigcap\_\; \backslash left(\backslash tfrac,\; 1-\backslash tfrac\backslash right]\; =\; \backslash bigcup\_n\; \backslash left(\backslash tfrac,\; 1\; -\; \backslash tfrac\backslash right]\; =\; (0,\; 1)$$and$$\backslash limsup\_\; A\_n\; =\; \backslash bigcap\_n\; \backslash bigcup\_\; \backslash left(\backslash tfrac,\; 1\; -\; \backslash tfrac\backslash right]\; =\; \backslash bigcap\_n\; \backslash left(-\backslash tfrac,\; 1\; +\; \backslash tfrac\backslash right]\; =\; [0,\; 1],$$so \lim _{n} A _{n} does not exist, despite the fact that the left and right endpoints of the intervals converge to 0 and 1, respectively.

• Let

A _{n} = \left\{ 0 , \tfrac{ 1 }{n}, \tfrac{ 2 }{n}, \ldots, \tfrac{n - 1 }{n}, 1 \right\}. Then$$\backslash bigcup\_\; A\_j\; =\; \backslash Q\backslash cap[0,1]$$is the set of all rational numbers between 0 and 1 (inclusive), since even for j < n and 0 \leq k \leq j, \tfrac{k}{j} = \tfrac{nk}{nj} is an element of the above. Therefore,$$\backslash limsup\_\; A\_n\; =\; \backslash Q\; \backslash cap\; [0,\; 1].$$On the other hand, $$\backslash bigcap\_\; A\_j\; =\; \backslash ,$$ which implies$$\backslash liminf\_\; A\_n\; =\; \backslash .$$In this case, the sequence A _{1,} A _{2,} \ldots does not have a limit. Note that \lim _{n} A _{n} 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,

(X, l{F}, P ) is a probability space, which means l{F} is a σ-algebra of subsets of X and P is a probability measure 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 l{F} then \lim _{n} A _{n} exists and$$\backslash mathbb\backslash left(\backslash lim\_\; A\_n\backslash right)\; =\; \backslash lim\_\; \backslash mathbb\backslash left(A\_n\backslash right).$$

Borel–Cantelli lemmas

See main article: article and Borel–Cantelli lemma. 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 isThe 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

Y _{1,} Y _{2,} \ldots converges to another random variable Y is formally expressed as $\backslash left\backslash .$ It would be a mistake, however, to write this simply as a limsup of events. That is, this the event $\backslash limsup\_\; \backslash left\backslash$! Instead, the of the event is Therefore,$$\backslash mathbb\backslash left(\backslash left\backslash \backslash right)\; =\; \backslash lim\_\; \backslash mathbb\backslash left(\backslash limsup\_\; \backslash left\backslash \backslash right).$$

Notes and References

1. Book: Resnick. Sidney I.. A Probability Path. 1998. Birkhäuser. Boston. 3-7643-4055-X.
2. Book: Gut, Allan . Probability: A Graduate Course: A Graduate Course . 2013 . Springer New York . 978-1-4614-4707-8 . Springer Texts in Statistics . 75 . New York, NY . en . 10.1007/978-1-4614-4708-5.