Measure Theory and the Central Limit Theorem

Transcription

1 Measure Theory ad the Cetral Limit Theorem Peter Brow August 24, 2011 Abstract It has log bee kow that the stadard iema itegral, ecoutered first as oe of the most importat cocepts of calculus, has may shortcomigs, ad ca oly be defied over the limited class of sets which have boudaries of Jorda cotet zero. I this paper I preset the measure theory ecessary to develop the more robust Lebesgue itegral. This is iterestig i its ow right, ad although we dwell little o the cosequeces of our defiitio, ay calculus studet who is familiar with the properties of the iema itegral will recogize its tremedous advatages. But we will push further: measure theory ad the Lebesgue itegral provide us with the ideal backgroud to develop a rigorous theory of probability. To this ed, we itroduce radom variables ad develop the theory of distributio fuctios. This paper culmiates i the proof of the Cetral Limit Theorem (CLT), which explais the ubiquitous ature of the ormal distributio. This is a deep ad fasciatig result, but relatively straightforward oce oe has provided the correct machiery. Ideed, we do most of the hard work studyig measure theory. This paper does ot provide the most ituitive approach towards the CLT, but it has the advatage of providig a itroductio to rigorous probability that may serve as a startig poit for future study. We have tried to provide the most expediet treatmet of the material possible without sacrificig the ituitio or the depth of the theoretical backgroud. 1 Itroductio This paper is desiged to provide a expositio of basic measure theory ad the Lebesgue itegral i preparatio for puttig probability theory o a rigorous foudatio ad provig the Cetral Limit Theorem. It is primarily iteded for the reader who has ever bee exposed to these cocepts before, ad as such we sped much time givig defiitios ad provig basic facts. esults such as the Fubii Theorem are proved i great detail so that readers ew to this topic may see how the large body of machiery we have developed works i practice. Ideed, all the material we preset is ecessary to uderstad the proof of the Cetral Limit Theorem, which is the fial goal of this paper. It is our hope that the reader the recogizes that the primary coectio betwee the first several sectios of the paper ad the last is ot merely philosophical: we eed measure 1

2 theory to formalize distributio fuctios, ad we eed itegratio theory to formalize characteristic fuctios. The first sectio attempts to motivate the study of the objects of measure theory, particularly σ-algebras, measurable fuctios, ad measures. The secod defies the Lebesgue itegral ad provides proofs of some of its most importat properties. We prove the Lebesgue Mootoe Covergece Theorem ad the Domiated Covergece Theorem. Much of the expositio of this sectio ad some additioal remarks are devoted to comparig the iema ad Lebesgue itegrals. The third sectio is devoted etirely to provig the Fubii Theorem. This may appear to be a strage choice. The proof of Fubii is log ad techical much more so tha its equivalet i a calculus course. However, there are importat reasos for presetig this material. eadig ad uderstadig the proofs of this sectio require a strog uderstadig of the material i the previous sectios ad, as such, provides the reader with a barometer of his or her uderstadig of the topic. Further, the Fubii Theorem is a highly importat ad useful computatioal result which will frequetly be of use to us. The fourth sectio is devoted to defiig the basic fuctioal tools of probability theory. We itroduce radom variables, expectatios, distributio fuctios, ad characteristic fuctios ad prove umerous basic results about each. While the vocabulary is familiar to ayoe who has had a itroductio to probability, the sectio is etirely measure-theoretic. Sectio five proves the Lévy Cotiuity Theorem, which is the tool eeded to prove the Cetral Limit Theorem, ad the fial sectio is devoted to the Cetral Limit Theorem itself. 2 Measure Theory Let be a oempty set. How are we to measure the size of a subset S? Naïvely, oe might cout the umber of elemets of S. Alteratively, if is edowed with a metric, we could try to defie the size of a set i terms of the set s diameter. These methods are ot particularly subtle, ad either is capable of, for example, determiig the dimesio of a set. The easiest coclusio is that our questio is ill-posed: oe eeds to first determie what properties the otio of size should have. Ideally, we would have some fuctio mappig from P() [0, ], which takes a subset of to a oegative umber (or ifiity). It seems reasoable to demad that this fuctio be additive across fiitely may disjoit sets, ad assig 0 to the empty set, ad after that we might work out additioal properties from a more detailed examiatio of ; for example, i the case of we might decide that every oempty ope iterval should have positive size. Hece, the fuctio that assigs to a iterval its legth seems a good startig poit for our ituitio. There is at least oe major difficulty with this process. I may cases, icludig the example above, it is ot possible to defie a fuctio o the etire power set of which has all the properties we might wat. This is ot obvious a priori, ad we will ot have the time to delve ito this topic deeply. Istead, we begi artificially by defiig the sorts of subcollectios of P() o which more restricted measures of size make sese. Philosophical questios 2

3 focusig o why we choose these particular axioms may be raised i the cotext of probability theory (particularly i geeralizig probability theory to certai aspects of Quatum Mechaics presetly a active field of research), but we will ot cocer ourselves with these either. We begi with a defiitio. Defiitio 2.1. A algebra of sets is a collectio F of subsets of some set, i.e. F P(), such that the followig coditios are satisfied: i., F. ii. E F, E c \ E F. iii. {E i } N i1 F, we have N i1 E i F. We say that is a σ-algebra if it additioally satisfies: iv. {E i } i N F, we have i1 F. Elemets of F are called measurable. I other words, we say that F is a σ-algebra if it is a algebra which is closed uder coutable uios. The axioms for a σ-algebra are relatively strog. Ay σ-algebra also satisfies the axioms for a topology. This is a apt compariso because a topology also gives us a sese of size of sets. Ituitively, a ope set is big, a closed set is small, ad a compact set is a geeralizatio of a fiite set. Usually a σ-algebra is too large to be iterestig as a topology, but sometimes it will be useful to cotrast these two structures. emark 2.2. We call F a algebra because (F,,,, ) is a algebra over the field with two elemets, where is symmetric differece ad ad are respectively the additive ad multiplicative uits of the algebra. Actually, it is also a field. Some texts, for example [3], refer to algebras of sets as fields ad σ-algebras of sets as σ-fields, ad this termiology is commo i probability. A σ-algebra is also closed uder coutable itersectios. If, agai, {E i } i N is a collectio of subsets, the i E i i (Ec i )c ( i Ec i )c by DeMorga s Law, ad this last is cotaied i the σ-algebra sice the σ-algebra is closed uder complemetatio ad coutable uio. Furthermore, ay σ-algebra F o iduces a σ-algebra o E. I fact, it is easy to see that E P(E) F satisfies the axioms for a σ-algebra o E. (i.) ad (iv.) follow trivially, ad we eed oly check that for all A E we have E \ A E, where E \ A E A c. Sice E, A F we have E\A E A c F by the previous remark. Takig this to its logical coclusio, it is easy to verify that the itersectio of two σ-algebras is agai a σ-algebra. Hece, as we do i may other cases i mathematics, we ca defie the σ-algebra geerated by some collectio of sets C by A C i I A i where {A i } is the collectio of all σ-algebras cotaiig C. Defiitio 2.3. We call a pair (, F), where is a oempty set ad F is a σ-algebra o, a measurable space. 3

4 I the cotext of topology, oe defies a cotiuous fuctio by requirig that a cotiuous map f : iduce a map f 1 : T ( ) T (). As we have oted, the cocept of a measurable space is similar to that of a topological space. Hece it seems reasoable to study fuctios that preserve the same type of structure. We will start by examiig real-valued fuctios, where the relevat σ-algebra o is the so-called Borel algebra geerated by the euclidea topology o. Defiitio 2.4. We say that a fuctio f : (, F) is measurable if for all α, we have f 1 [(α, )] F. We deote the space of measurable realvalued fuctios o by M(, F), ad the space of o-egative measurable real-valued fuctios o by M(, F) +. This is the sort of measurable fuctio we will be most cocered with, but it is ot the most geeral otio we could have give. If is aother measurable space, or more geerally a topological space, we might say that a fuctio f : is measurable if for every measurable, respectively ope set U i, that f 1 (U) belogs to F. It is also useful to kow if measurability is preserved uder certai operatios, such as compositio, additio, ad so o. The followig propositio aswers these questios. We use the more geeral defiitio of a measurable fuctio because this serves to simplify our otatio. Propositio 2.5. Let (, F) be a measurable space. i. If ad Z are topological spaces, f : is measurable, ad g : Z is cotiuous, the h : g f is measurable. ii. If is a topological space, u, v : are measurable fuctios, ad Φ : 2 is cotiuous, the h(x) Φ(u(x), v(x)) is measurable. iii. If f :, g : are measurable ad η the f + g, fg, ad ηf are also measurable fuctios. Proof. To prove i., let U Z be a ope set. The h 1 (U) (f g) 1 (U) f 1 (g 1 (U)). V g 1 (U) is ope by cotiuity, so f 1 (V ) is measurable. To prove ii., it suffices to show that f(x) (u(x), v(x)) is measurable by i. Note that all ope sets i 2 are uios of ope rectagles, hece it suffices to show that for ay ope rectagles (a, b) (c, d), f 1 [(a, b) (c, d)] is a measurable set. But f 1 [(a, b) (c, d)] u 1 [(a, b)] v 1 [(c, d)] which is measurable. Fially, sice Φ(x, y) x + y, Φ(x, y) xy, ad Φ(x, y) ηx are cotiuous fuctios, iii. follows from ii. We will also frequetly be iterested i cosiderig the limit of a sequece of fuctios, or the limit supremum ad limit ifimum. We briefly recall these 4

5 otios for geeral sequeces. Oe way to thik of the limit ifimum ad the limit supremum of a sequece a is as the ifimum ad supremum respectively of the set S {x : x is a accumulatio poit of {a }}. However, it ca be hard to work directly from this defiitio, so we give a equivalet oe. Defiitio 2.6. The limit ifimum ad limit supremum of a sequece {a }, deoted lim if a ad lim sup a, are defied by ad lim if a sup lim sup a if if k a k sup a k. If a sequece has a limit, the lim a lim sup a lim if a. We leave this, ad the equivalece of the two defiitios give, as exercises for the reader. Propositio 2.7. Let {f } be a sequece of measurable fuctios. The φ(x) if f (x), Φ(x) sup f (x), f(x) lim if f (x), ad f(x) lim sup f (x) are measurable fuctios. Proof. First, we claim that Φ 1 [(α, )] f 1 [(α, )]. Clearly we must have f 1 [(α, )] Φ 1 [(α, )]. To see the other iclusio, assume we have x such that Φ(x) (α, ) but f (x) α for all N. The clearly Φ(x) sup f (x) α, so that Φ(x) / (α, ) ad the two sets are equal. Note each elemet i the uio f 1 [(α, )] is measurable, ad the uio is coutable; hece it is a measurable set by the defiitio of σ-algebra. Thus Φ is a measurable fuctio. The argumet for φ is aalogous. Next, recall that we ca write f(x) if sup k f k (x) ad f(x) sup if k f k (x). These are the compositios of the fuctios Φ ad φ with coutably may members of the sequece {f }, hece also measurable. Up util ow we have bee describig collectios of measurable sets ad fuctios, ad particularly what combiatios of these objects we would hope remai measurable. We have yet to actually defie a fuctio that measures them. As oted i the expositio, we expect that this fuctio will have certai ice properties. For example, give two disjoit sets A ad B, the measure of A B ought to be the measure of A plus the measure of B. (I later sectios, whe we tur our focus to probability, the property we have just described is very atural. If we idetify measurable sets with possible evets ad the fuctio which measures them with their probability of occurrig, this property implies that if A has probability P 1 of occurrig ad B has probability P 2 ad the two evets are mutually exclusive, the A B has probability P 1 + P 2 of occurrig). The followig defiitios clarify this ituitio. Defiitio 2.8. A measure space is a triple (, F, µ) where (, F) is a measurable space ad µ : F [, ] is a fuctio satisfyig the followig: i. µ( ) 0. k 5

6 ii. E, µ(e) 0. iii. If {A } is a collectio of pairwise disjoit sets, the µ( A ) µ(a ). We call the size fuctio µ a measure. If µ has properties i. ad iii. but ot ii. we call µ a siged measure. If µ is a measure that additioally satisfies iv. µ() 1, the µ is referred to as a probability measure ad we refer to µ(a) as the probability that A occurs. I this case we refer to the elemets of F as evets, ad to as a probability space. Probability measures are special cases of measures where we restrict the values of µ(e) to be fiite; these fuctios are called fiite measures. It is a easy corollary of the ext propositio that a measure µ is a fiite measure if ad oly if µ() <. If there exists a coutable coverig of our space of sets i F such that all sets i the coverig have fiite measure, the we say µ is σ-fiite. Our ext propositio outlies some basic properties of measure. Propositio 2.9. Let (, F, µ) be a measure space. Let A, B, {E } F; the the followig properties hold: i. if A B the µ(a) µ(b). ii. if A B ad µ(a) + the µ(b \ A) µ(b) µ(a). iii. if {E } is a sequece of icreasig sets, the µ( E ) lim µ(e ). iv. if µ(e 1 ) + ad {E } is decreasig, the µ( E ) lim µ(e ). Proof. First ote that fiite additivity of µ follows from coutable additivity (property iii. of the defiitio of measure). Now, if A B the sets A, B \ A are disjoit. Sice the measure of ay set i F is oegative we have µ(a) µ(a) + µ(b \ A) µ(a (B \ A)) µ(b); this proves i. Furthermore, ii. follows from the secod to last equality. Let {E } be a icreasig sequece; that is, E E +1 for all. Note that {E +1 \ E } is a sequece of disjoit sets, where we covetioally set E 1. Note also that E E +1 \ E. Thus, by coutable additivity, µ( E ) µ( E +1 \ E ) µ(e +1 \ E ) µ(e +1) µ(e ) by ii., but this is a telescopig series. Sice N µ( E ) 1 N [µ(e +1 ) µ(e )] µ(e N ), 1 6

7 we have lim N µ(e N ) lim N µ( N 1 E ) µ( E ). Now suppose {E } is a decreasig sequece. Defie A E 1 \ E for each. Note that {A } is a icreasig sequece; the µ( A ) µ( E 1 \ E ) µ(e 1 \ E ) µ(e 1 ) µ( E ), ad also µ( A ) lim µ(a ) lim µ(e 1 \E ) lim µ(e 1 ) µ(e ) µ(e 1 ) lim µ(e ) by (iii). Equatig these expressios ad subtractig µ(e 1 ) gives iv. Oce we have chose a σ-algebra F ad a measure µ over a space we ca begi to phrase results i terms of the size of subsets of. Ofte sets of larger size will be more importat tha sets of smaller size. For example, sets of measure zero that is, sets N F such that µ(n) 0 are i some sese egligible. I fact, i may cases we may fid that a statemet holds everywhere except over such a set of measure 0. If this is the case, we say this property holds almost everywhere; if is a probability space we say the property occurs almost surely. May theorems require oly that we assume our hypotheses hold almost everywhere. Coversely, if we are workig i a fiite measure space we regard sets of full measure as extremely importat. 3 Itegratio The itegral is a atural geeralizatio of the cocept of a sum. Geerally oe s first itroductio to this geeralizatio which exteds the cocept of summatio to the cotiuum is the fairly atural iema itegral. If f is a sufficietly ice fuctio defied o some iterval [a, b], a iema sum cosists of a fiite set of terms of the form f(x i )(x i+1 x i ), where the x j are a ordered partitio of [a, b] ad x i [x i, x i+1 ]. Each term approximates some of the area uder the curve of f. ecallig our discussio of what properties a size fuctio should have, we ote that oe reasoable way to defie the size of a ope iterval of is as its legth; ideed, if oe sets µ([a, b)) b a, there is a caoical extesio of µ from the set of all fiite uios ad itersectios of disjoit half-ope itervals to the Borel algebra o. The resultig measure is called Lebesgue measure. The term (x i+1 x i ) above is exactly the Lebesgue measure of the iterval [x i, x i+1 ), ad the iema sum does ot use ay of the field properties of the real umbers. Hece we might hope to rephrase the defiitio of itegral to hold for real-valued fuctios o a arbitrary measure space. We will fid that this ca be doe, ad that the resultig object o, called the Lebesgue itegral, is more robust tha the iema itegral. It behaves better with respect to limits of sequeces of fuctios ad exteds the rage of the stadard iema itegral sigificatly. However, these seredipitous cosequeces are ot obvious a priori. 7

8 Defiitio 3.1. Let (, F, µ) be a measure space. A simple fuctio is a measurable fuctio ϕ : which takes oly a fiite umber of distict values. That is, ϕ is a simple fuctio if A {a : ϕ(x) a for some x } is a fiite set. We ca always represet a simple fuctio i the form ϕ i1 a i1i Ei where the a i are costats, the E i are measurable sets, ad 1I B is the idicator fuctio of the set B. If we order A as a subset of, we might set E i ϕ 1 ({a i }) where A {a i } i1, ad this gives the caoical represetatio for ϕ. We will almost always work with this represetatio, but oe of our results deped o this choice. The reader may check that the followig defiitio is idepedet of the represetatio chose for ϕ. Defiitio 3.2. If ϕ i1 a i1i Ei is a oegative simple fuctio, the ϕ dµ a i µ(e i ) i1 is called the itegral of ϕ with respect to µ. The ext propositio shows that this ew itegral is liear ad shows how we ca produce a ew measure by takig the itegral of a set E as its measure. Propositio 3.3. Let (, F, µ) be a measure space. If ϕ ad ψ are oegative simple fuctios ad η + the the followig hold: i. ϕ + ψ ad ηφ are oegative simple fuctios. ii. ηϕ dµ η ϕ dµ iii. iv. ϕ + ψ dµ ϕ dµ + ψ dµ ϕχ E dµ λ(e) defies a measure o (, F). Proof. It is clear that ϕ + ψ ad ηϕ are oegative simple fuctios. Propositio 2.5 tells us they are also measurable. Let ϕ i1 a i1i Ai be the caoical represetatio for ϕ; the by defiitio, ηϕ dµ ηa i µ(a i ) η i1 i1 a i µ(a i ) η ϕ dµ. It is ot difficult to check that if ψ m j1 b j1i Bj is the caoical represetatio for ψ, we the have ϕ + ψ i1 j1 m (a i + b j )1I Ai B j. 8

9 Further, ote that the A i s are pairwise disjoit as are the B j s sice we are dealig with the caoical represetatio. The m ϕ + ψ dµ (a i + b j )µ(a i B j ) i1 j1 m a i µ(a i B j ) + i1 j1 m a i i1 j1 i1 µ(a i B j ) + m a i µ(a i ) + b j µ(b j ) j1 ϕ dµ + ψ dµ, m b j µ(a i B j ) i1 j1 m b j j1 i1 µ(a i B j ) where we have used oly basic properties of measures ad fiite sums. That λ( ) 0 ad λ(e) 0 follow from the defiitio of the itegral of a simple fuctio. It remais to show coutable additivity. Suppose E i is a coutable family of pairwise disjoit sets. The If we allow we have ϕ1i i Ei dµ j1 α j µ(a i i i Ei ϕ dµ ϕ E i ) α j 1I Aj j1 ϕ1i i Ei dµ j1 α j µ( i A j E i ) α j µ(a j E i ) Where we have applied the coutable additivity of µ. Note that the first sum is fiite so we ca certaily chage the order of summatio. The we have α j µ(a j E i ) ϕ dµ i j i E i j1 i Of course i some sese this result is ot very iterestig because we have ot yet exteded the itegral to geeral measurable fuctios. Let us first defie the more geeral itegral for positive fuctios. We will the exted the itegral to ay measurable fuctio by makig use of the fact that f f + f where f + ad f are defied by f + (x) { f(x) if f(x) > 0, 0 otherwise; f (x) { f(x) if f(x) < 0,, 0 otherwise 9

10 ad each of the above is a positive fuctio. Defiitio 3.4. Let (, F, µ) be a measure space, ad let M(, F) + {f : : f is F-measurable ad f(x) 0 x } ad Φ + f {ϕ M(, F)+ : ϕ is simple ad 0 ϕ(x) f(x) x }. If f M(, F) +, we defie the itegral of f with respect to µ to be f dµ sup ϕ dµ ϕ Φ + f Moreover, we defie the itegral of f over ay measurable set E by f dµ f1i E dµ E Now we tur our attetio to the iteractio of the itegral with limits. It is well kow that give a sequece of fuctios {f } which are iema itegrable with limit f, the limit of the itegrals eed oly coverge to the itegral of f provided the covergece of f to f is uiform. But we shall see that there are several weaker criteria which imply this type of covergece for Lebesgue itegrals. These results are some of the most importat improvemets of the Lebesgue itegral over the stadard iema itegral. The first criterio is mootoicity of the sequece of measurable fuctios {f }. Theorem 3.5. (Mootoe Covergece Theorem) Let (, F, µ) be a measure space, ad let {f } be a icreasig sequece of fuctios i M(, F) + covergig poitwise to a fuctio f :. The f M(, F) + ad we have: f dµ lim f dµ Proof. Note that sice lim f (x) f(x) exists, we have lim sup f (x) lim f (x) f(x), which is measurable by propositio. Further, by hypothesis f (x) 0 for all x, so that f(x) 0 for all x, ad f M(, F) +. Sice f (x) f(x), it follows that f dµ is a bouded mootoically icreasig sequece ad i particular has a limit sice f dµ (the boud) exists; this shows half of what we wat to prove. It remais to show that we also have the opposite iequality. Fix a oegative simple fuctio ϕ(x) where ϕ(x) f(x) for all x. We first cosider the fuctio αϕ where α (0, 1) ad defie E E (α) {x : αϕ(x) f (x)}. 10

11 Now, sice f (x) coverges poitwise to f(x) ad ϕ(x) f(x) by assumptio, we have, for ay x, either αϕ(x) f(x) or αϕ(x) f(x) 0, i which case we also have f (x) 0 for all because f is oegative ad bouded above by f. Hece i either case we ca fid some N such that ϕ(x) f N (x) f +1 (x), so that E. (This was our motivatio for itroductig the costat α; we caot make this claim without strict iequality). The, for ay, αϕ dµ α ϕ dµ f dµ f dµ E E E This tells us that α lim E ϕ dµ lim f dµ Propositio 2.14 ad the fial part of Propositio 3.3 tell us that lim ϕ dµ ϕ dµ E I particular, we have sup α lim ϕ dµ sup α α (0,1) E α (0,1) ϕ dµ lim f dµ But the, sice this procedure works for ay simple fuctio such that ϕ f we have f dµ sup ϕ dµ lim f dµ φ Φ + f emark 3.6. I our proof of the Mootoe Covergece Theorem we used the fact that if ϕ is a oegative fuctio ad E F the E ϕ dµ F ϕ dµ. Theorem 3.7. Let (, F, µ) be a measure space. If ϕ ad ψ are oegative measurable fuctios ad η + the the followig hold: i. ϕ + ψ ad ηφ are oegative measurable fuctios. ii. ηϕ dµ η ϕ dµ. iii. (ϕ + ψ) dµ ϕ dµ + ψ dµ. iv. If ϕ ψ the ϕ dµ ψ dµ. v. λ(e) E ϕ dµ defies a measure o (, F). 11

12 Proof. As i the proof of 3.3, i. is clear. Next, choose a mootoically icreasig sequece of oegative simple fuctios {ϕ } covergig to ϕ ad a similar sequece {ψ } covergig to ψ ad ote that {ηϕ } ad {ϕ + ψ } are mootoically icreasig sequeces of oegative simple fuctios covergig to ηϕ ad ϕ + ψ respectively. Applyig Propositio 3.3, we the ivoke the Mootoe Covergece Theorem to see lim ηϕ dµ η lim ϕ dµ η ϕ dµ ad lim ϕ + ψ dµ lim ϕ dµ + lim ψ dµ ϕ dµ + ψ dµ, provig ii. If we ow suppose that ϕ ψ the by assumptio ψ ψ ϕ. The usig the otatio of Defiitio 3.4 we have ψ dµ sup g dµ g Φ + f sice ψ Φ + f. iii. the follows from the Mootoe Covergece Theorem. Fially, certaily λ( ) 0. For all E F, we have ϕ1i E 0, hece λ(e) 0. Let {E i } be a coutable collectio of pairwise disjoit sets. Let 1I k k i1 1I E i. Clearly lim k 1I k 1I ad {1I k} is a mootoically icreasig sequece. Note i Ei that {ϕ1i k } is also a mootoically icreasig sequece covergig to ϕ1i Ei. i Hece, λ( E i ) ϕ1i dµ lim ϕ1i i Ei k dµ, k i ad this last is exactly lim ϕ 1I Ei dµ lim k k i1 k ϕ1i Ei dµ ϕ1i Ei dµ i i i1 where we have agai applied the Mootoe Covergece Theorem. λ(e i ), This together with our precedig remark establishes the liearity of the geeral itegral. Note that this methodology removes the messy ad uelighteig maipulatio of iema sums required to show the iema itegral is liear. I this cotext, liearity appears as a cosequece of the way i which the Lebesgue itegral behaves with respect to limits, whereas i some sese the proof of this fact for iema itegrals obfuscates what is actually goig o. I fact, takig the measure-theoretic poit of view has o dowside: it is trivial to show that the Lebesgue itegral is a extesio of the iema itegral. That is, ay iema itegrable fuctio is Lebesgue itegrable ad their itegrals coicide. We do ot prove this result here because we have ot actually costructed the Lebesgue measure o, oly argued that such a measure if existet meets the ituitive requiremets for a size fuctio o the real lie. 12

13 Defiitio 3.8. Let (, F, µ) be a measure space ad let L(, F, µ) be the set {f M(, F) such that f + ad f have a fiite itegral with respect to µ}, where (as we metioed before) f + ad f are defied by { { f + f(x) if f(x) > 0, (x) f f(x) if f(x) < 0, (x). 0 otherwise; 0 otherwise For all f L(, F, µ) the itegral of f with respect to µ is defied to be f dµ f + dµ f dµ We fiish our itroductio to measure theory by examiig some other ways i which the itegral iteracts icely with limits. Propositio 3.9. (Fatou s Lemma) Let (, F, µ) be a measure space. Let {f } be a sequece of fuctios i M(, ) + ; the f dµ lim if f dµ. Proof. The fuctio if k f k (x) is measurable by Propositio 2.7. if k f k (x) f k (x) for k. This implies if f k dµ f k dµ k Clearly for k. I particular this remais true whe we take the ifimum over k. Thus lim if f k dµ lim if f k dµ; k k but ote that if k f k (x) is a sequece of mootoically icreasig positive fuctios idexed by. Hece by the Mootoe Covergece Theorem lim if f k dµ lim if f k dµ lim if f k dµ. k k k But by the defiitio of the limit ifimum this is what we wated to show. Propositio If f M(, F), the i. f L(, F, µ) if ad oly if f L(, F, µ). ii. f dµ f dµ. 13

14 Proof. Assume f L(, F, µ). The f + f f + + f which are measurable ad have fiite itegrals by assumptio, ad f 0. Thus clearly f L(, F, µ). O the other had, if f L(, F, µ) ote that f + f ad f f sice f f f. The clearly these itegrals are fiite by compariso; hece f L(, F, µ). This proves i. ii. follows sice f f f implies f dµ f dµ f dµ. emark Note that the previous Propositio is ot true for the iema itegral. Cosider, for example { 1 if x Q f(x) 1 if x \ Q o the iterval [0, 1]. f, beig a costat, is iema itegrable, but f, beig discotiuous everywhere, is ot. Theorem (Domiated Covergece Theorem) Let (, F, µ) be a measure space ad {f } a sequece of real fuctios i M(, F) covergig to a real valued fuctio f M(, F). Let g L(, F, µ) be a oegative fuctio. If f g, the f, f L(, F, µ) ad f dµ lim f dµ Proof. First ote that f g so the itegral of f is fiite, hece f L(, F, µ) sice g is. The by Propositio 1.6 we have f L(, F, µ). It follows that f L(, F, µ) also sice f dµ lim f dµ f dµ lim if f dµ which is clearly fiite sice each f has a fiite itegral. Sice g ± f 0 we apply Fatou s lemma to {g ± f } This yields lim if(g ± f ) dµ g dµ ± lim if f dµ g dµ ± f dµ lim if g ± f dµ ( ) lim if g dµ ± f dµ Cosequetly, g dµ + f dµ g dµ + lim if f dµ 14

15 ad g dµ f dµ g dµ + lim if( f dµ) g dµ lim sup f dµ Subtractig the commo term from all of these iequalities ad multiplyig the secod by egative oe yields lim sup f dµ f dµ lim if f dµ Hece lim f dµ f dµ ad i particular exists. 4 The Fubii Theorem We have developed the machiery for defiig ad calculatig itegrals over abstract spaces, but the theory so far does ot take ito accout ay additioal structure o those spaces. For example, though our theory allows us to itegrate over we must, so to speak, itegrate all at oce. We would like to treat as..., ad the itegrate over by separately itegratig over each copy of. Towards this ed we will ow describe how the cartesia product of two measure spaces may be regarded as a measure space. First we assume that we have two measure spaces (,, µ) ad (,, ν). If A ad B are measurable we call A B a measurable rectagle. We are iterested i the σ-algebra geerated by the set of all measurable rectagles. Defiitio 4.1. We deote the σ-algebra geerated by the set of measurable rectagles by. The associated measurable space is (, ). We refer to this as the cartesia product of the measurable spaces (, ) ad (, ). We would like to defie a measure π o (, ) which is i some way compatible with the measures o (, ) ad (, ). Oe compatibility coditio we might expect is that, for ay measurable rectagle A B, we should have π(a B) µ(a)ν(b). This would be i accordace with the usual way we compute areas of geometric objects i 2. Ideed, defiig π this way gives a measure o the set of fiite uios of measurable rectagles, which is a algebra, but this set is ot i geeral a σ-algebra. It is ot obvious a priori that we ca exted π to be a measure o, or that there should be a uique such extesio, ad provig this requires some additioal machiery, i particular the Carathéodory Extesio Theorem, the Hah Extesio Theorem, ad the costructio of outer measure. These are iterestig topics, but ot of vital importace to our developmet. I lieu of developig this machiery, we will simply state the theorem that we will require i the sequel. 15

16 Theorem 4.2. (Product Measure) If (,, µ) ad (,, ν) are measure spaces, the there exists a measure π : such that π(a B) µ(a)ν(b) for every measurable rectagle A B. If µ, ν are σ-fiite, the π is uique ad σ-fiite. For a more complete developmet of these topics see [1]. The reader may recall from multivariable calculus the Fubii Theorem, which is itroduced as a meas of computig double itegrals by iteratio; this is exactly the treatmet we described i the itroductio to this chapter. I preparatio for statig the more geeral measure-theoretic Fubii Theorem we eed to cosider objects that are i some sese oe dimesioal slices of the sets i our product measure space. This motivates the followig defiitio. Defiitio 4.3. Let (, ) be as i Defiitio 4.1. Let S. The if x, y we defie the x-sectio of S by ad the y-sectio of S by S x {y : (x, y) S} S y {x : (x, y) S} I a similar maer, we ca defie sectios of fuctios. Defiitio 4.4. Let (, ) be as i Defiitio 4.1, ad let f be a real-valued measurable fuctio o. The if x, y we defie f x : ad f y : by f x (y) f(x, y) ad f y (x) f(x, y) ad call f x the x-sectio of f ad f y the y-sectio of f. Our goal is to develop a aalogue of the Fubii theorem by cosiderig sectios of a product set S. We will itegrate over all the, say, x-sectios of a set with respect to the measure ν associated to. Oe would hope that with suitable measurability coditios that this process would recover the full measure of the set S. Propositio 4.5. Let (, ) be as i Defiitio 4.1. Let S be measurable. The: i. S y ad S x for every x ad y. ii. If f : is -measurable the f y is -measurable ad f x is -measurable. Proof. For i. it suffices to show that the sets ( ) x {E : E x, x } 16

17 ad ( ) y {E : E y, y } are both σ-algebras cotaiig all measurable rectagles. The it follows immediately that both must cotai because this is the σ-algebra geerated by the set of measurable rectagles. We prove that ( ) x is a σ-algebra ad the proof for the other set is the same. First ote that for x, ( ) x ad also x. Now, it is easy to check that, if E ( ) x the (( ) \ E) x ( ) x \ E x sice is a σ-algebra. Now let E ( ) x. It is also easy to check that ( ) E (E ) x x because by assumptio (E ) x by assumptio. The ( ) x is a σ-algebra ad hece it must cotai. It is easy to check that ad (f y ) 1 [(, a)] (f 1 [(, a)]) y (f x ) 1 [(, a)] (f 1 [(, a)]) x Both of these sets are measurable by part i. sice f is a measurable fuctio, so f y, f x are measurable also. We are almost ready to prove the Fubii Theorem. techical lemma, which we give below. We eed oe more Defiitio 4.6. A mootoe class is a o-empty collectio C of subsets of some set such that i. For every icreasig sequece {E } C, we have E C. ii. For every decreasig sequece {E } C, we have E C. Propositio 4.7. (Mootoe Class Lemma) Let A be a algebra of sets. The the mootoe class C(A) geerated by A is the same as the σ-algebra σ(a) geerated by A. Proof. The proof is basically formal ad is omitted. Oe starts by showig that the itersectio of mootoe classes is agai a mootoe class, ad that every σ-algebra is a mootoe class. Corollary 4.8. Let C be a mootoe class cotaiig a algebra A. The C cotais the σ-algebra geerated by A. We ow have all the tools we will eed to prove our lemma. This ext result is essetially the Fubii Theorem for characteristic fuctios. Oce we have this result provig the more geeral Fubii Theorem is straightforward. 17

18 Lemma 4.9. Let (,, µ) ad (,, ν) be σ-fiite measure spaces. The for ay E the fuctios f E : ad g E : give by are measurable ad f E (x) ν(e x )ad g E (y) µ(e y ) f E dµ π(e) g E dν Proof. We begi by provig this theorem assumig our measure spaces are fiite. The we will exted the result to the class of σ-fiite spaces. Suppose P is the algebra of sets geerated by measurable rectagles. Set S {f E M(, ) +, g E M(, ) +, ad f E dµ π(e) g E dµ}. We wat to show that S is a σ-algebra cotaiig P. If we succeed i this, because is geerated by P, we will have show that S cotais. It is easy to see that this set cotais every measurable rectagle A B. Ideed, ote that for ay x, y we have f E µ(e y ) µ(a)1i B (y) ad similarly g E ν(e x ) ν(b)1i A (x). Hece f E dµ ν(b)1i A dµ ν(b) 1I A dµ µ(a)ν(b), ad tracig the argumet backwards this is also equal to g E dν. Now for ay set P P we ca write P i A i B i as the uio of disjoit rectagles. Cosequetly, if π deotes the measure o as before, we have π(p ) π( i A i B i ) i i π(a i B i ) ν(b i )1I Ai dµ ν(b i )1I Ai dµ i µ((a i B i ) y ) dµ, i ad also µ((a i B i ) y ) µ( i i (A i B i ) y ) µ(( i A i B i ) y ) 18

19 sice the sets above are disjoit ad it is simple to check that the uio of y- sectios is the y-sectio of the uio. We also used the Mootoe Covergece Theorem i the secod-to-last equality of the array above. Thus π(p ) µ(( A i B i ) y ) dµ f dµ f i Ai Bi P dµ. i A similar argumet establishes the equality π(p ) g P dµ, ad we deduce P S. Now, it remais to show that S is a mootoe class; we will the apply the Mootoe Class Lemma to fiish the proof. To this ed, let {E } be a icreasig sequece of sets i S. For all we have f E dµ π(e ) g E dν. Now for ay x ad y, the sequeces {(E ) x } ad {(E ) y } are also icreasig. Clearly lim f E f E ad lim g E g E, ad furthermore these are mootoically icreasig sequeces. Also, lim π(e ) π( E ) π(e). Hece, applyig the Mootoe Covergece Theorem we have lim f E dµ f E dµ lim g E dµ g E dµ lim π(e ) π(e). Similarly, assume {E } is a decreasig sequece. Agai it is clear that lim f E lim ν((e ) x ) ν( (E ) x ) ν(( E ) x ) ν(e) f E ad similarly lim g E g E. Agai, lim π(e ) lim π( E ) π(e). Here our sequece of fuctios is ot icreasig, but ote that 0 f E K ad 0 g E K for K max{µ(), ν( )} which exists sice our measure spaces are fiite. Thus, we may apply the Domiated Covergece Theorem ad after takig limits we deduce f E dµ π(e) g E dν. Hece S is a mootoe class cotaiig P, ad thus cotais. Now let us assume our measure spaces are σ-fiite but ot ecessarily fiite. Clearly the product measure space is also. Let {A } be a measurable coverig of such that for all N, A A +1 ad π(a ) <. For each we defie a ew fiite measure space o the σ-algebra P(A ). We deote 19

20 µ or ν restricted to this space by µ ad ν respectively. Let E be a measurable set i (, ) ad defie E E A. First ote that lim f E (x) lim ν((e A ) x ) lim ν(e x (A ) x ) ν( E x (A ) x ) ν(e x ( A ) ν(e x ) ν(e x ) f E (x) Where we have used the fact that {A } is a icreasig sequece. Similarly, lim g E (y) g E (y). Further, it is obvious that these sequeces are icreasig. Sice P(A ) is a fiite measure space we have f E dµ f E dµ 1I E dπ g E dν g E dν where we have used the fact that sice E A, µ ad µ coicide, as do ν ad ν. Hece, the Mootoe Covergece Theorem tells us that takig the limit of this expressio yields f E dµ dπ g E dν E Now we will prove the Fubii Theorem for o-egative fuctios. Theorem (Fubii-Toelli) Let (,, µ) ad (,, ν) be σ-fiite measure spaces. If h M(, ) + the f h ad g h give by f h (x) h x dν; g h (y) h y dµ belog respectively to M(, ) + ad M(, ) + ad f h dµ h dπ g h dν. Proof. First, let us assume h 1I E for some measurable set E. It follows from the defiitio of sectio that (1I E ) x 1I Ex ad (1I E ) y 1I Ey. But the, we have f h (x) ν(e x ) ad g h (y) µ(e x ). Hece Lemma 4.9 gives the desired result immediately. Liearity of the itegral the exteds this result to all positive measurable simple fuctios. Fially, assume h M(, ) + ad let {h } be a sequece of mootoically icreasig positive simple fuctios covergig to h. It is easy to verify that {(h ) x } ad {(h ) y } are also sequeces of mootoically icreasig positive simple fuctios. Furthermore, basic properties of the itegral show that {f h } ad {g h } are mootoically icreasig sequeces of simple fuctio as well. The ubiquitous Mootoe Covergece Theorem applied to {(h ) x } ad {(h ) y } the implies lim f h f h ad lim g h g h. Fially, sice for all we have f h dµ h dπ g h dν, 20

21 takig the limit as goes to ifiity ad applyig the Mootoe Covergece Theorem completes the proof of the geeral statemet. Theorem (Fubii) Let (,, µ) ad (,, ν) be σ-fiite measure spaces. If h L(,, π) the the fuctios f h : ad g h : defied almost everywhere by f h (x) h x dν ad g h (y) h y dµ belog respectively to L(,, µ) ad L(,, ν), ad f h dµ h dπ g h dν. Proof. The fuctios h + ad h satisfy the hypothesis of Theorem Thus, we have f h + dµ h + dπ g h + dν ad f h dµ Combiig these gives (f h + f h ) dµ But ote that f h + f h (h + ) x (h ) x dν h dπ (h + h ) dπ g h dν. (h x ) + (h x ) dν (g h + g h ) dν. h x dν f h (x). The aalagous equality for g h also holds; all we have used here is the fact that (h ± ) x (h x ) ±, which follows at oce from their defiitios. Hece the above equality may be writte as f h dµ h dπ g h dν, as desired. emark Actually, this proof ivolved a bit of had-wavig i that we glossed over a uillumiatig subtlety. We do ot a priori kow that f h + ad f h are everywhere fiite. However, we do kow that they have fiite itegrals by the Toelli Theorem. Cosequetly, there may be a set N of measure zero o which our above argumet does ot hold. That is, we may ot be able to sesibly write f h + f h o the etire space. I practice oe simply excises the set N from discussio ad igores it, which is why i the statemet of the theorem we oted that f h ad g h were defied almost everywhere. 21

22 The Fubii Theorem, while perhaps ot vital for uderstadig the Cetral Limit Theorem, will be importat to us later whe dealig with idicator fuctios. It is iterestig to ote that provig Fubii for the iema itegral is somewhat trivial, but the theorem for the Lebesgue itegral requires much more machiery. However, our approach based aroud developig the otio of sectios seems to provide more isight tha the estimatio of upper ad lower sums that is used i the iema case, mostly because our assumptios are much less restrictive tha i the iema case. We required oly that h be itegrable, whereas for the iema case oe much check that h is itegrable ad that all the iterated itegrals exist. The reaso we do ot require these assumptios is because they are roughly aalogous to the results of Propositio 4.5, which followed from the fact that h is measurable. 5 Distributios ad Idicator Fuctios Now we wish to develop the basic cocepts of probability theory. We already kow what a probability space is - it is a measure space with total mass 1. We ofte deote such a space by Ω, ad a elemet of such a space ω Ω, called a outcome. Measurable subsets of Ω are called evets, ad measurable fuctios o a probability space are called radom variables. We itroduce a probabilistic way of thikig about these fuctios, ad the itroduce distributio fuctios, first as derived from radom variables ad the as totally separate quatities. Fially we itroduce idicator fuctios as a meas of workig with distributio fuctios. The reaso for this approach will become clear i the ext sectio, where we will eed the theory of idicator fuctios to prove the Cetral Limit Theorem. Defiitio 5.1. If (Ω, F, P) is a probability space, the a radom variable is a measurable fuctio : Ω. If dp < the we defie the expectatio, expected value, or mea of to be E() dp. We also briefly defie oe more cocept which we will require later. Defiitio 5.2. The variace of a radom variable is give by Var() : σ() 2 E( E()). It is easy to check that σ() 2 E( 2 ) E() 2. Ofte, the square root of the variace, σ σ(), is referred to as the stadard deviatio of. The simplest example of a radom variable is the idicator fuctio 1I A that we have already used frequetly We call this the characteristic fuctio of the set A ad deote it χ A whe we are thikig of our space as a measure space. 22

23 We may ecouter a case where we have two spaces, (Ω, F, P) ad (Ω 1, F 1 ), ad a measurable fuctio h : Ω Ω 1. The we ca defie a measure ν h o (Ω 1, F 1 ) by ν h (A 1 ) P[h 1 (A 1 )]. Furthermore, if f 1 is a radom variable o (Ω 1, F 1, ν h ) the f(x) f 1 [h(x)] is a radom variable o (Ω, F, P) ad they have the same expected value. Propositio 5.3. If either side exists, the E(f) f dp f 1 dν h Proof. We prove this first for the idicator fuctio 1I A : Ω 1. There (1I A h) dp P[h 1 (A)] ν h (A) 1I A dν h. Now we cosider the case of a simple fuctio ϕ : Ω 1 give by ϕ(x) α i 1I Ai (x) i1 where α i ad A i Ω 1. I this case, (ϕ h) dp i1 α i (1I Ai h) dp α i (1I Ai h) dp i1 α i 1I Ai dν h i1 i1 ϕ dν h, α i 1I Ai dν h by liearity of the itegral. Fially, let f 1 : Ω 1 be a arbitrary measurable fuctio. The, as usual, we ca choose a sequece of mootoically icreasig simple fuctios {f 1 } which coverge to f 1. The sequece f f 1 h also icreases mootoically, ad coverges to f. Hece, by the Mootoe Covergece Theorem we have E(f) f dp lim f dp lim f 1 dν h lim f 1 dν h which fiishes the proof. We are ofte iterested i a radom variable ad the probability that it takes values i various sets with respect to a probability measure µ. 23

24 Defiitio 5.4. Let (Ω, F, P) be a probability space ad a radom variable. We defie the probability distributio or law of to be the fuctio P (B) µ( 1 (B)) for B. We defie the distributio fuctio of to be the fuctio F (x) P ((, x]) We have the obvious cosequece that for a < b we have F (b) F (a) µ{(a, b]}. Corollary 5.5. If g : the we have E[g ] (g ) dp Ω g dp. Proof. This is Propositio 5.3 applied to the machiery of Defiitio 5.4. Propositio 5.6. Let be a radom variable ad F be its distributio fuctio. The F has the followig properties i. F is o-decreasig ii. F is cotiuous from the right iii. iv. lim F (x) 1 x lim F (x) 0 x Proof. i. follows immedietely sice x y implies (, x) (, y) which shows that 1 [(, x)] 1 [(, y)] so i particular F (x) P( 1 [(, x)]) P( 1 [(, y)]) F (y) ii. follows sice if {x } is a sequece covergig to x from the right lim F (x ) lim P( 1 [(, x )] x x+ but { 1 [(, x )]} is a decreasig sequece, hece by Propositio 2.9 ( ) lim P( 1 [(, x )]) P 1 [(, x )] So F is right cotiuous. iii. ad iv. are similar ad are left to the reader. ( ]) [ P 1 (, x ) P ( 1 (, x] ) F (x) 24

25 emark 5.7. This proof used that fact that if x coverges to x from the right the (, x ) (, x]. Note that if x coverges to x from the left but x x for ay, the (, x ) (, x). This subtle poit is the reaso a distributio fuctio may fail to be cotiuous. Coversely, it is ofte the case that for a fuctio satisfyig some of the above properties that it is the distributio of some radom variable. This is ot always the case, but it motivates the followig more geeral defiitio. Defiitio 5.8. A fuctio F : is said to be a distributio fuctio if i. 0 F 1. ii. F is o-decreasig. iii. F is cotiuous from the right. We defie the variatio of a distributio fuctio by Var(F ) lim F (x) lim F (x). x x If F is the distributio fuctio of a radom variable we have Var(F ) 1. We use the same otatio for variatio ad variace, but o cofusio should result as the settig will always make it clear which cocept is iteded. Defiitio 5.9. If F is a distributio fuctio, we defie the characteristic fuctio φ of F by φ(t) φ F (t) e itx df, where this itegral is take with respect to the measure iduced by the distributio fuctio, as described i the prelude to Propositio 5.3. emark Propositio 5.3 also implies that φ(t) e itx df e itf dp. We have ot provided ay motivatio for such a object; however, it will tur out to be very useful because, as we shall prove i a momet, the characteristic fuctio uiquely defies the distributio fuctio. The characteristic fuctio is a iterestig object of study ot oly because it is equivalet to the distributio fuctio i this sese, but also because it is frequetly easier to work with. Propositio Let be a radom variable with characteristic fuctio φ. The i. For α, β, we have φ (α+β) (t) e iβt φ(αt). Ω 25

26 ii. If E( ) < the d dt φ (t) t0 i E( ). iii. If f is a arbitrary distributio fuctio, the for ay t, φ(t) Var(f); furthermore φ(0) Var(f). I particular, if f F is the distributio fuctio of a radom variable, the φ(t) 1 ad φ(0) 1. Proof. To prove i., we have φ (α+β) (t) e it(α+β) dp e i(αt) e iβt dp e iβt e iαtx df (x) Next, e iβt φ(αt). d d φ(t) dt dt e itx df (x), ad it is a cosequece of the Domiated Covergece Theorem that we may iterchage the derivative ad the itegral. The reader may see [3] for the precise statemet of the ecessary propositio. Thus we have d dt eitx df (x) (ix) e itx df (x), ad hece Fially, φ(t) d dt φ(t) t0 (ix) df (x) i E( ). e itx df (x) Sice e 0 1, this is a equality for φ(0). df (x) lim F (x) x lim x 6 Covergece of Distributio Fuctios F (x) Var(F ). This sectio s fuctio is primarily techical. Our aim is to determie whe covergece i characteristic fuctio implies covergece i distributio. It is ot obvious yet why this will be of iterest to us. However, we will fid that the characteristic fuctio of a sum of radom variables is much easier to work with tha the correspodig distributio fuctio. 26

27 Defiitio 6.1. A sequece of distributio fuctios {F } is said to coverge weakly if F (x) F (x) for each x C(F ) {x : F is cotiuous at x}. It is said to coverge up to a additive costat to a distributio fuctio F if for all subsequeces {F k } ad {F j } such that F k coverges weakly to F 1 ad F j coverges weakly to F 2 we have F 1 F 2 c for some costat c. I most cases we will be dealig with cotiuous distributio fuctios ad C(F ). We iclude this requiremet oly for completeess. As we oted i the previous sectio it is possible for a distributio fuctio to be discotiuous. Distributio fuctios are ecessarily right cotiuous, but i strage situatios, such as whe certai sigletos have positive measure, cotiuity may fail. Settig aside these techical cosideratios, we begi our study of the covergece of distributio fuctios with the followig propositio. Propositio 6.2. Let {F } be a sequece of distributio fuctios. The there exists a subsequece F k covergig weakly to some distributio fuctio F. Proof. We make use of a diagoal argumet. Let {q i } be a eumeratio of the ratioal umbers. Cosider the sequece {F (q 1 )}. 0 F 1 by assumptio, so this is a bouded sequece of real umbers which by the Bolzao-Weierstrass Theorem has a coverget subsequece. Deote such a subsequece by {F i1 }. Now take a subsequece of this sequece such that {F ik 2 (q 2)} coverges ad label this sequece {F i2 }. Iterate this process to produce a diagoal sequece {F ii }. For sufficietly large i this is a subsequece of some {F kj } ad hece {F k j(q)} is coverget for each q ratioal. At all ratioal poits we defie F (x) lim i F ii (x). F is icreasig sice it is the limit of icreasig fuctios. Now, we defie F (x) if{f (q) : q Q, q x}. It is apparet from this defiitio that F, ow defied o, is icreasig ad right cotiuous. Sice F is bouded over the ratioals it must be bouded over the reals also because it is mootoically icreasig ad right cotiuous. Thus F is a distributio fuctio. Sometimes this result is referred to as the Helly Selectio Priciple. Now we tur our attetio to the characteristic fuctio oce agai. The followig otio will be useful to us later. Defiitio 6.3. Let F be a distributio fuctio with characteristic fuctio φ. The itegral characteristic fuctio φ for F is the fuctio φ(t) t 0 φ(v) dv. emark 6.4. With the setup of Defiitio 6.3, we have t t φ : φ(v) dv e ivx df (x) dv e ivx dv df (x)

28 by Fubii. Itegratig, we have φ(t) e itx 1 ix df (x) We will eed oe further result to aalyze covergece. emark 6.5. It is a stadard result from aalysis that if {a } is a sequece of real umbers, the a a if ad oly if for every subsequece a k there is a further (sub)subsequece a kj which coverges to a. This strage characterizatio of covergece will be useful for provig the ext propositio. Propositio 6.6. i. Let {F } ad F be distributio fuctios with itegral characteristic fuctios {φ } ad φ, respectively. The if F F up to a additive costat, we also have φ φ poitwise over. ii. If φ g poitwise over the there is a distributio fuctio F with itegral characteristic fuctio φ such that F F up to a additive costat ad g φ. Proof. Suppose that F F up to a additive costat. Let F 1 ad F 2 be subsequeces such that F 1 F 1 ad F 2 F 2. Followig emark 6.4, sice e itx 1 ix 2 x, which is bouded ad cotiuous away from 0, we see that as x ± e itx 1 ix 0 Give this coditio, it is a cosequece of a versio of the Helley-Bray Lemma, which may be foud i chaper 8 of [3], that e itx 1 ix df 1 (x) e itx 1 ix df 1 (x) ad the aalogous result is true for F 2. Thus φ 1 φ 1 ad φ 2 φ 2. We have by assumptio F 1 F 2 c for some c which implies that the measures iduced by these fuctios are the same ad the same as that iduced by F, hece e itx 1 ix df 1 (x) e itx 1 ix df 2 (x) This shows that give ay coverget subsequece of F, the correspodig subsequece of φ coverges, ad the limits of each subsequece coicide. Now, pickig a arbitrary subsequece F k, there is a further subsequece F kl which does i fact coverge by Propositio 6.2. Further, the limits of all such subsequeces differ by oly a additive costat. The above argumet shows that give ay subsequece φ k there is a further subsequece φ kl which coverges, 28