Sigma Field Notes for Economics 770 : Econometric Theory

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1 1 Sigma Field Notes for Economics 770 : Econometric Theory Jonathan B. Hill Dept. of Economics, University of North Carolina 0. DEFINITIONS AND CONVENTIONS The following is a small list of repeatedly used symbols and terms with de nitions. Consult any probability text book, including Davidson (1994) and Fristedt and Gray (1997), or any listed on the syllabus, for larger lists. 0.1 Notation/Symbol Conventions F is an algebra (in this document, it is almost always a -algebra/ eld) N = f1; 2; :::g the space of positive integers Q is the space of rational numbers Z = f::: 2; 1:0; 1; 2; :::g the space of integers is the sample space! is an event, or element A; B; C are subsets of ; fa j g is a sequence of sets. A c is the complement of A: A c = =A = A A=B = A B = f! 2 A :! 2 A \ B c g : the set A with elements removed if they are also in B fx n g is a sequence of real numbers. 0.2 Mathematical De nitions Countable: If there exists a one-to-one mapping to (a subset of) the integers N (e.g. f1; 2g; N; Q). inf j fa j g = \ 1 j=1 A j: the largest set contained in each A j (it may be empty?) lim inf n!1 x n = lim n!1 inf mn x n = sup n1 inf mn x n ; hence lim inf n!1 A n = [ 1 n=1f\ 1 m=na n g lim sup n!1 x n = lim n!1 sup mn x n = inf n1 sup mn x n ; hence lim sup n!1 A n = \ 1 n=1f[ 1 m=na n g Partition of A: A collection of disjoint subsets fb j g of A such that [ 1 j=1 B j = A. sup j fa j g = [ 1 j=1 A j: the smallest set containing all A j (it may be the sample space ) Uncountable: Countable does not hold. Any open or closed real interval (e.g. (a; b] for any a < b).

2 2 1. SAMPLE SPACE, FIELDS, BOREL FIELDS Consult Bierens (chapt. 1.1, ) for details. See also Davidson (chapt. 1). Any textbook on measure theory will also be helpful. 1.1 Statistical experiment, sample space, events! DEFN. Statistical Experiment: An activity with at least one possible outcome; the set of possible outcomes is known; the outcome itself has an element of chance. DEFN. Sample Space: The set of all possible outcomes or events!: EXAMPLE (sample space): The experiment is to ip a coin once. Possible outcomes are H = heads or T = tails. = fh; T g. EXAMPLE (sample space): The experiment is to ip a coin n times and count the number of heads, denoted X. Possible outcomes are X = 0; 1; 2; :::; n heads. = f0; 1; :::; ng. 1.2 Field, -Field The sample is not rich enough to describe all possible event combinations that may arise from. Although in Example there are n + 1 possible outcomes, there are many more random events that can be described. Let n = 5. The event that at least 3 heads occur: X 2 f3; 4g. Clearly that is also a random event, but it is not a possible outcome contained in, although it is a subset of : Algebras, in particular -algebras, give us su cient richness of event possibilities. DEFN. Algebra/Field: An algebra or eld F is a collection of subsets of with the following properties: i. If A 2 F then A c 2 F (it is closed under compliments) ii. If A j 2 F for j = 1; :::; n then [ n j=1 A j 2 F (closed under nite unions) Thus F is a fairly rich set of subsets of collections of events, but not rich enough for the types of probabilistic problems we face. If countably in nite unique subsets exist then Property (ii) clearly omits some cases. The appropriate richness follows if any countable union is in F. DEFN. -Algebra or -Field: A eld F is a - eld if it is closed under countable unions: 1. If A 2 F then A c 2 F (it is closed under compliments) 2. If A j 2 F for j = 1; :::; 1 then [ 1 j=1 A j 2 F (closed under countable unions). COMMENT: We will use - elds to describe the possible array of events associated with statistical experiments and therefore with random variables X. A random variable X will be de ned only relative to some space of outcomes, and some - eld F of richly collected event subsets that can fully describe the values that X can take. If an - eld is closed under compliments and countable unions then it must contain the sample space : if A 2 F then A c ; A c [ A = 2 F, hence c =? 2 F. A - eld. F contains? and. Thus, we can de ne the - eld with the least degree of richness. DEFN. Trivial -algebra or trivial - eld: F = f?; g.

3 The de nitions of eld and - eld are slightly misleading. In particular, they are not unique (in general) to a particular, and some have more structure than others. EXAMPLE (- eld): Flip a coin three times and count the number of heads. = f0; 1; 2; 3g. Then F = f?; ; f0; 1; 2g; f3gg is a - eld. If any A; B 2 F then A c 2 F, A [ B 2 F, and so on. We will see below that F is not su ciently rich to describe this particular random experiment. EXAMPLE (- eld): Flip a coin twice and count the number of heads. = f0; 1; 2g. Then F = f?; ; f0g; f1g; f2g; f0; 1g; f0; 2g; f1; 2gg is a - eld. 1.3 Properties of - elds If is a nite set then there can only be nitely many countable unions: [ 1 j=1 A j must involve redundant sets, i.e. [ 1 j=1 A j = [ n j=1 A j for some n. Let F be an eld on nite. F is a - eld. EXAMPLE ( nite space): Roll a die. = f1; 2; 3; 4; 5; 6g. Any eld F of subsets of is a - eld. Consider F = f?; ; f2; 4; 6g; f1; 3; 5gg. By de Morgan s law if [ 1 j=1 A j 2 F then ( [ 1 j=1 A j) c = \ 1 j=1 Ac j 2 F. Since A j can be anything (another set s compliment, for example!) it follows a - eld is closed under countable intersections. The same goes for a eld. EXAMPLE (sigma- eld spanning sets): Suppose has a partition A 1 and A 2, that is A 1 [ A 2 =. Then fa 1 ; A 2 g is not necessarily a - eld, except in very simple cases. We can easy create a - eld by 1. adding compliments, and 2. adding countable unions. Then F := (fa 1 ; A 2 g) = f?; ; A 1 ; A 2 ; A c 1; A c 2; A c 1 [A c 2; (A c 1 [A c 2) c g is a - eld of subsets of : if B i 2 F then B c i 2 F and [ ib i 2 F are easily veri ed. Algebras and - elds are closed under countable intersections. The above examples are so primitive as to be useless for most economists most of the time. The entire point is to build intuition to the point that we can comfortably understand how all probability matters reduce to measures of - elds. The following moves us into the right direction. First, intersections of even uncountably many - elds is a - eld, but the property does not necessarily apply to unions. Let F, 2, be a collection of - elds of subsets of, where may be an uncountable set (e.g. a compact subset of R). Then F := \ 2 F is a - eld. PROOF: If fa j g 1 j=1, each A j 2 F, then each A j is in every F by the de nition of an intersection. Therefore each A c j and [1 j=1 A j is in every F since they are - elds. Therefore A c j 2 F and [1 j=1 A j 2 F. The same applies to and?. QED. EXAMPLE (unions of - elds may not be a - eld): [ 2 F need not be closed under countable unions. Take = [0; 1] and F n = f?; ; [0; 1 1=n]; (1 1=n; 1]g for n 1. Then F n is a - eld, and A n = [0; 1 1=n] 2 F n 2 [ 1 n=1f n, but [ 1 n=1a n = [0; 1) is not in any F n and therefore cannot be in [ 1 n=1f n. The key idea here is the limit: an in nite union can contain a limiting object not present in any particular - eld so it cannot be in the union of them. 3

4 COMMENT: The latter example has the key implication that we cannot arbitrarily "join" - elds and assume we indeed still have a - eld. In most cases of interest (i.e. not rolling a die!) we work with very abstract notions of "events" so although we cannot write "" we need not know its contents other than "!". What exactly is the total set of events! that drive an interest rate? or the decision to buy life insurance? Yet we must and will join - elds of subsets of events on because we will inspect many variables at once (e.g. an interest rate and the unemployment rate). So, beware! Verifying if a collection of subsets F is a - eld may be di cult. In particular, showing countable unions lie in F may be quite challenging. One trick is to show monotone sequences of sets fa n g in F have a limit lim n!1 A n in F. The latter property de nes a monotone class. DEFN. Monotone Class F: A class of sets F, such that if fa n g is a monotone sequence and A n 2 F 8n, then lim n!1 A n 2 F. F is a - eld if and only if it is an eld and a monotone class. 4 Think about it: - elds have sets, their compliments, their nite unions, and of course much more (extending to countable unions). So, if you show F is a eld, and show increasingly larger sets all A n 2 F and also lim n!1 A n 2 F then you will have covered countably in nite unions which are implicitly contained in those monotone sequences. 1.4 Generate/Smallest - eld The preceding lemma will be useful for de ning the smallest possible - eld containing some subset of interest C. DEFN. - eld generated by C, or (C): The smallest - eld containing a collection C of sets. Finding a - eld that contains C is easy since we need only add unions and compliments to "complete" C. But this does not necessarily lead to a unique - eld. If ff g 2 is the collection of all - elds that contains C, then the smallest - eld containing C is identically (C) := \ 2 F : EXAMPLE (- eld generated by R): Let C = f( 1; q]; q 2 Qg, the collection of halflines with ration endpoints. Since Q is countable, so is C. Then B := (C) is the smallest - eld containing C. This is by de nition the Borel eld of R. COMMENT: The - eld generated by a collection of subsets is profoundly important. In practice we are most interested in - elds generated by a random variable x, de ned below. That is, we want to know the minimal collection of subsets of events that are related to x. Once we know this we can measure the likelihood of x. EXAMPLE (- eld generated): Roll a die hence = f1; 2; 3; 4; 5; 6g. Let C = f1; 3; 5g. The smallest - eld containing C is simply (C) = f?; ; f1; 3; 5g; f2; 4; 6g : This is the - eld generated by an odd die roll. It is also the - eld generated by an even die roll (verify). 1.5 Borel sets B B

5 As we saw above, a Borel eld is a special kind of generated - eld. It is generated by the intervals on the real line. DEFN. Borel Field B: The - eld generated by C = f(a; b) : a < b, a; b 2 Rg, the set of open intervals. This is denoted as B := (C) An element of B os called a Borel set. Borel elds are not unique: collections of [a; b] or ( 1; a], or ( 1; q] for rational q 2 Q, all generate the same B. B = (( 1; a] : a 2 Rg). This is the collection of half-lines, their unions and compliments and intersections. Such pure generality lends itself to measuring, or capturing the statistical properties, of mappings X :! R (which we will call a random variable), and their transformations g(x). 5