1Sigma 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 denitions. Consultany probability text book, including Davidson (1994) and Fristedt and Gray (1997), or any listedon 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 integersQ is the space of rational numbersZ = f::: 2;1:0; 1; 2; :::g the space of integers

is the sample space! is an event, or elementA;B;C are subsets of ; fAjg is a sequence of sets.Ac is the complement of A: Ac = =A = AA=B = AB = f! 2 A : ! 2 A\Bcg : the set A with elements removed if they are also in Bfxng is a sequence of real numbers.

0.2 Mathematical Denitions

Countable: If there exists a one-to-one mapping to (a subset of) the integers N (e.g. f1; 2g; N;Q).infjfAjg = \1j=1Aj : the largest set contained in each Aj (it may be empty ?)lim infn!1 xn = limn!1 infmn xn = supn1 infmn xn; hence lim infn!1An = [1n=1f\1m=nAnglim supn!1 xn = limn!1 supmn xn = infn1 supmn xn; hence lim supn!1An =\1n=1f[1m=nAngPartition of A: A collection of disjoint subsets fBjg of A such that [1j=1Bj = A.supjfAjg = [1j=1Aj : the smallest set containing all Aj (it may be the sample space )Uncountable: Countable does not hold. Any open or closed real interval (e.g. (a; b] for anya < b).

21. SAMPLE SPACE, FIELDS, BOREL FIELDS

Consult Bierens (chapt. 1.1, 1.3-1.4) for details. See also Davidson (chapt. 1). Any textbook onmeasure 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 possibleoutcomes is known; the outcome itself has an element of chance.

DEFN. Sample Space: The set of all possible outcomes or events !:

EXAMPLE 1.1.1 (sample space): The experiment is to ip a coin once. Possible outcomes areH = heads or T = tails. = fH;Tg.EXAMPLE 1.1.2 (sample space): The experiment is to ip a coin n times and count thenumber 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 1.1.2 there are n + 1 possible outcomes, there are many more randomevents that can be described.

Let n = 5. The event that at least 3 heads occur: X 2 f3; 4g. Clearly that is also a randomevent, but it is not a possible outcome contained in , although it is a subset of : Algebras, inparticular -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 followingproperties:

i. If A 2 F then Ac 2 F (it is closed under compliments)ii. If Aj 2 F for j = 1; :::; n then [nj=1Aj 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 innite unique subsets exist then Property (ii) clearlyomits 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 Ac 2 F (it is closed under compliments)2. If Aj 2 F for j = 1; :::;1 then [1j=1Aj 2 F (closed under countable unions).

COMMENT: We will use -elds to describe the possible array of events associated withstatistical experiments and therefore with random variables X. A random variable X will bedened only relative to some space of outcomes, and some -eld F of richly collected eventsubsets 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 samplespace : if A 2 F then Ac; Ac [A = 2 F , hence c = ? 2 F .LEMMA: A -eld. F contains ? and .

Thus, we can dene the -eld with the least degree of richness.

DEFN. Trivial -algebra or trivial -eld : F = f?;g.

3The denitions 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 1.2.1 (-eld): Flip a coin three times and count the number of heads. =f0; 1; 2; 3g. Then

F = f?;; f0; 1; 2g; f3ggis a -eld. If any A;B 2 F then Ac 2 F , A [B 2 F , and so on. We will see below that F is notsu ciently rich to describe this particular random experiment.

EXAMPLE 1.2.2 (-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; 2ggis a -eld.

1.3 Properties of -elds

If is a nite set then there can only be nitely many countable unions: [1j=1Aj must involveredundant sets, i.e. [1j=1Aj = [nj=1Aj for some n.LEMMA: Let F be an eld on nite . F is a -eld.EXAMPLE 1.3.1 (nite space): Roll a die. = f1; 2; 3; 4; 5; 6g. Any eld F of subsets of isa -eld. Consider F = f?;; f2; 4; 6g; f1; 3; 5gg.

By de Morgans law if [1j=1Aj 2 F then ( [1j=1Aj)c = \1j=1Acj 2 F . Since Aj can be anything(another sets compliment, for example!) it follows a -eld is closed under countable intersections.The same goes for a eld.

EXAMPLE 1.3.2 (sigma-eld spanning sets): Suppose has a partition A1 and A2, that isA1 [A2 = . Then fA1; A2g 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. ThenF := (fA1; A2g) = f?;; A1; A2; Ac1; Ac2; Ac1[Ac2; (Ac1[Ac2)cg is a -eld of subsets of : if Bi 2 Fthen Bci 2 F and [iBi 2 F are easily veried.LEMMA: 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 allprobability 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.

LEMMA: Let F, 2 , be a collection of -elds of subsets of , where may be anuncountable set (e.g. a compact subset of R). Then F := \2F is a -eld.PROOF: If fAjg1j=1, each Aj 2 F , then each Aj is in every F by the denition of anintersection. Therefore each Acj and [1j=1Aj is in every F since they are -elds. ThereforeAcj 2 F and [1j=1Aj 2 F . The same applies to and ?. QED.EXAMPLE 1.3.3 (unions of -elds may not be a -eld): [2F need not be closedunder countable unions. Take = [0; 1] and Fn = f?;; [0; 1 1=n]; (1 1=n; 1]g for n 1. ThenFn is a -eld, and An = [0; 1 1=n] 2 Fn 2 [1n=1Fn, but [1n=1An = [0; 1) is not in any Fn andtherefore cannot be in [1n=1Fn.

The key idea here is the limit : an innite union can contain a limiting object not present inany particular -eld so it cannot be in the union of them.

4COMMENT: 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 knowits 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, showingcountable unions lie in F may be quite challenging. One trick is to show monotone sequences ofsets fAng in F have a limit limn!1An in F . The latter property denes a monotone class.DEFN. Monotone Class F : A class of sets F , such that if fAng is a monotone sequence andAn 2 F 8n, then limn!1An 2 F .LEMMA: F is a -eld if and only if it is an eld and a monotone class.

Think about it: -elds have sets, their compliments, their nite unions, and of course muchmore (extending to countable unions). So, if you show F is a eld, and show increasingly larger setsall An 2 F and also limn!1An 2 F then you will have covered countably innite unions whichare implicitly contained in those monotone sequences.

1.4 Generate/Smallest -eld

The preceding lemma will be useful for dening the smallest possible -eld containing somesubset 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 fFg2 is the collectionof all -elds that contains C, then the smallest -eld containing C is identically

(C) := \2F:

EXAMPLE 1.4.1 (-eld generated by R): Let C = f(1; q]; q 2 Qg, the collection of half-lines with ration endpoints. Since Q is countable, so is C. Then B := (C) is the smallest -eldcontaining C. This is by denition the Borel eld of R.

COMMENT: The -eld generated by a collection of subsets is profoundly important. Inpractice we are most interested in -elds generated by a random variable x, dened below. Thatis, we want to know the minimal collection of subsets of events that are related to x. Once we knowthis we can measure the likelihood of x.

EXAMPLE 1.4.2 (-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

5As we saw above, a Borel eld is a special kind of generated -eld. It is generated by theintervals 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 openintervals. 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, allgenerate the same B.LEMMA: B = ((1; a] : a 2 Rg).

This is the collection of half-lines, their unions and compliments and intersections. Such puregenerality 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).