Preface to First Edition
This book in measure-theoretic probability has resulted from classroom lecture notes that this author has developed over a number of years, by teaching such a course at both the University of Wisconsin, Madison, and the University of California, Davis. The audience consisted of graduate students primarily in statistics and mathematics. There were always some students from engineering departments, and a handful of students from other disciplines such as economics.
The book is not a comprehensive treatment of probability, nor is it meant to be one. Rather, it is an excursion in measure-theoretic probability with the objective of introducing the student to the basic tools in measure theory and probability as they are commonly used in statistics, mathematics, and other areas employing this kind of moderately advanced mathematical machinery. Furthermore, it must be emphasized that the approach adopted here is entirely classical. Thus, characteristic functions are a tool employed extensively; no use of martingale or empirical process techniques is made anywhere.
The book does not commence with probabilistic concepts, and there is a good reason for it. As many of those engaged in teaching advanced probability and statistical theory know, very few students, if any, have been exposed to a measure theory course prior to attempting a course in advanced probability. This has been invariably the experience of this author throughout the years. This very fact necessitates the study of the basic measure-theoretic concepts and results—in particular, the study of those concepts and results that apply immediately to probability, and also in the form and shape they are used in probability.
On the basis of such considerations, the framework of the material to be dealt with is therefore determined. It consists of a brief introduction to measure theory, and then the discussion of those probability results that constitute the backbone of the subject matter. There is minimal flexibility allowed, and that is exploited in the form of the final chapter of the book. From many interesting and important candidate topics, this author has chosen to present a brief discussion of some basic concepts and results of ergodic theory.
From the very outset, there is one point that must be abundantly clarified, and that is the fact that everything is discussed in great detail with all proofs included; no room is allowed for summary unproven statements. This approach has at least two side benefits, as this author sees them. One is that students have at their disposal a comprehensive and detailed proof of what are often deep theorems. Second, the instructor may skip the reproduction of such proofs by assigning their study to students.
In the experience of this author, there are no topics in this book which can be omitted, except perhaps for the final chapter. With this in mind, the material can be taught in two quarters, and perhaps even in one semester with appropriate calibration of the rate of presentation, and the omission of proofs of judiciously selected
theorems. With all details presented, one can also cover an entire year of instruction, perhaps with some supplementation.
Most chapters are supplied with examples, and all chapters are concluded with a varying number of exercises. An unusual feature here is that an Answers Manual of all exercises will be made available to those instructors who adopt the book as the textbook in their course. Furthermore, an overview of each one of the 15 chapters is included in an appendix to the main body of the book. It is believed that the reader will benefit significantly by reviewing the overview of a chapter before the material in the chapter itself is discussed.
The remainder of this preface is devoted to a brief presentation of the material discussed in the 15 chapters of the book, chapter-by-chapter.
Chapter 1 commences with the introduction of the important classes of sets in an abstract space, which are those of a field, a σ-field, including the Borel σ-field, and a monotone class. They are illustrated by concrete examples, and their relationships are studied. Product spaces are also introduced, and some basic results are established. The discussion proceeds with the introduction of the concept of measurable functions, and in particular of random vectors and random variables. Some related results are also presented. This chapter is concluded with a fundamental theorem, Theorem 17, which provides for pointwise approximation of any random variable by a sequence of so-called simple random variables.
Chapter 2 is devoted to the introduction of the concept of a measure, and the study of the most basic results associated with it. Although a field is the class over which a measure can be defined in an intuitively satisfying manner, it is a σ-field—the one generated by an underlying field—on which a measure must be defined. One way of carrying out the construction of a measure on a σ-field is to use as a tool the so-called outer measure. The concept of an outer measure is then introduced, and some of its properties are studied in the second section of the chapter. Thus, starting with a measure on a field, utilizing the associated outer measure and the powerful Carathéodory extension theorem, one ensures the definition of a measure over the σ-field generated by the underlying field. The chapter is concluded with a study of the relationship between a measure over the Borel σ-field in the real line and certain point functions. A measure always determines a class of point functions, which are nondecreasing and right-continuous. The important thing, however, is that each such point function uniquely determines a measure on the Borel σ-field.
In Chapter 3, sequences of random variables are considered, and two basic kinds of convergences are introduced. One of them is the almost everywhere convergence, and the other is convergence in measure. The former convergence is essentially the familiar pointwise convergence, whereas convergence in measure is a mode of convergence not occurring in a calculus course. A precise expression of the set of pointwise convergence is established, which is used for formulating necessary and sufficient conditions for almost everywhere convergence. Convergence in measure is weaker than almost everywhere convergence, and the latter implies the former for finite measures. Almost everywhere convergence and mutual almost everywhere
convergence are equivalent, as is easily seen. Although the same is true when convergence in measure is involved, its justification is fairly complicated and requires the introduction of the concept of almost uniform convergence. Actually, a substantial part of the chapter is devoted in proving the equivalence just stated. In closing, it is to be mentioned that, in the presence of a probability measure, almost everywhere convergence and convergence in measure become, respectively, almost sure convergence and convergence in probability
Chapter 4 is devoted to the introduction of the concept of the integral of a random variable with respect to a measure, and the proof of some fundamental properties of the integral. When the underlying measure is a probability measure, the integral of a random variable becomes its expectation. The procedure of defining the concept of the integral follows three steps. The integral is first defined for a simple random variable, then for a nonnegative random variable, and finally for any random variable, provided the last step produces a meaningful quantity. This chapter is concluded with a result, Theorem 13, which transforms integration of a function of a random variable on an abstract probability space into integration of a real-valued function defined on the real line with respect to a probability measure on the Borel σ-field, which is the probability distribution of the random variable involved.
Chapter 5 is the first chapter where much of what was derived in the previous chapters is put to work. This chapter provides results that in a real sense constitute the workhorse whenever convergence of integrals is concerned, or differentiability under an integral sign is called for, or interchange of the order of integration is required. Some of the relevant theorems here are known by names such as the Lebesgue Monotone Convergence Theorem, the Fatou–Lebesgue Theorem, the Dominated Convergence Theorem, and the Fubini Theorem. Suitable modifications of the basic theorems in the chapter cover many important cases of both theoretical and applied interest. This is also the appropriate point to mention that many properties involving integrals are established by following a standard methodology; namely, the property in question is first proved for indicator functions, then for nonnegative simple random variables, next for nonnegative random variables, and finally for any random variables. Each step in this process relies heavily on the previous step, and the Lebesgue Monotone Convergence Theorem plays a central role.
Chapter 6 is the next chapter in which results of great utilitarian value are established. These results include the standard inequalities (Hölder (Cauchy–Schwarz), Minkowski, cr, Jensen), and a combination of a probability/moment inequality, which produces the Markov and Tchebichev inequalities. A third kind of convergence—convergence in the rth mean—is also introduced and studied to a considerable extent. It is shown that convergence in the rth mean is equivalent to mutual convergence in the rth mean. Also, necessary and sufficient conditions for convergence in the rth mean are given. These conditions typically involve the concepts of uniform continuity and uniform integrability, which are important in their own right. It is an easy consequence of the Markov inequality that convergence in the rth mean
implies convergence in probability. No direct relation may be established between convergence in the rth mean and almost sure convergence.
In Chapter 7, the concept of absolute continuity of a measure relative to another measure is introduced, and the most important result from utilitarian viewpoint is derived; this is the Radon–Nikodym Theorem, Theorem 3. This theorem provides the representation of a dominated measure as the indefinite integral of a nonnegative random variable with respect to the dominating measure. Its corollary provides the justification for what is done routinely in statistics; namely, employing a probability density function in integration. The Radon–Nikodym Theorem follows easily from the Lebesgue Decomposition Theorem, which is a deep result, and this in turn is based on the Hahn–Jordan Decomposition Theorem. Although all these results are proved in great detail, this is an instance where an instructor may choose to give the outlines of the first two theorems, and assign to students the study of the details.
Chapter 8 revolves around the concept of distribution functions and their basic properties. These properties include the fact that a distribution function is uniquely determined by its values on a set that is dense in the real line, that the discontinuities, being jumps only, are countably many, and that every distribution function is uniquely decomposed into two distribution functions, one of which is a step function and the other a continuous function. Next, the concepts of weak and complete convergence of a sequence of distribution functions are introduced, and it is shown that a sequence of distribution functions is weakly compact. In the final section of the chapter, the so-called Helly–Bray type results are established. This means that sufficient conditions are given under which weak or complete convergence of a sequence of distribution functions implies convergence of the integrals of a function with respect to the underlying distribution functions.
The purpose of Chapter 9 is to introduce the concept of conditional expectation of a random variable in an abstract setting; the concept of conditional probability then follows as a special case. A first installment of basic properties of conditional expectations is presented, and then the discussion proceeds with the derivation of the conditional versions of the standard inequalities dealt with in Chapter 6. Conditional versions of some of the standard convergence theorems of Chapter 5 are also derived, and the chapter is concluded with the discussion of further properties of conditional expectations, and an application linking the abstract definition of conditional probability with its elementary definition.
In Chapter 10, the concept of independence is considered first for events and then for σ-fields and random variables. A number of interesting results are discussed, including the fact that real-valued (measurable) functions of independent random variables are independent random variables, and that the expectation of the product of independent random variables is the product of the individual expectations. However, the most substantial result in this chapter is the fact that factorization of the joint distribution function of a finite number of random variables implies independence of the random variables involved. This result is essentially based on the fact that σ-fields generated by independent fields are themselves independent.
Chapter 11 is devoted to characteristic functions, their basic properties, and their usage for probabilistic purposes. Once the concept of a characteristic function is defined, the fundamental result, referred to in the literature as the inversion formula, is established in a detailed manner, and several special cases are considered; also, the applicability of the formula is illustrated by means of two concrete examples. One of the main objectives in this chapter is that of establishing the Paul Lévy Continuity Theorem, thereby reducing the proof of weak convergence of a sequence of distribution functions to that of a sequence of characteristic functions, a problem much easier to deal with. This is done in Section 3, after a number of auxiliary results are first derived. The multidimensional version of the continuity theorem is essentially reduced to the one-dimensional case through the so-called Cramér–Wold device; this is done in Section 4. Convolution of two distribution functions and several related results are discussed in Section 5, whereas in the following section additional properties of characteristic functions are established. These properties include the expansion of a characteristic function in a Taylor-like formula around zero with a remainder given in three different forms. A direct application of this expansion produces the Weak Law of Large Numbers and the Central Limit Theorem. In Section 8, the significance of the moments of a random variable is dramatized by showing that, under certain conditions, these moments completely determine the distribution of the random variable through its characteristic function. The rigorous proof of the relevant theorem makes use of a number of results from complex analysis, which for convenient reference are cited in the final section of the chapter.
In the next two chapters—Chapters 12 and 13—what may be considered as the backbone of classical probability is taken up: namely, the study of the central limit problem is considered under two settings, one for centered random variables and one for noncentered random variables. In both cases, a triangular array of row-wise independent random variables is considered, and, under some general and weak conditions, the totality of limiting laws—in the sense of weak convergence—is obtained for the row sums. As a very special case, necessary and sufficient conditions are given for convergence to the normal law for both the centered and the noncentered case. In the former case, sets of simpler sufficient conditions are also given for convergence to the normal law, whereas in the latter case, necessary and sufficient conditions are given for convergence to the Poisson law. The Central Limit Theorem in its usual simple form and the convergence of binomial probabilities to Poisson probabilities are also derived as very special cases of general results.
The main objective of Chapter 14 is to present a complete discussion of the Kolmogorov Strong Law of Large Numbers. Before this can be attempted, a long series of other results must be established, the first of which is the Kolmogorov inequalities. The discussion proceeds with the presentation of sufficient conditions for a series of centered random variables to convergence almost surely, the Borel–Cantelli Lemma, the Borel Zero–One Criterion, and two analytical results known as the Toeplitz Lemma and the Kronecker Lemma. Still the discussion of another two results is needed—one being a weak partial version of the Kolmogorov Strong Law
of Large Numbers, and the other providing estimates of the expectation of a random variable in terms of sums of probabilities—before the Kolmogorov Strong Law of Large Numbers, Theorem 7, is stated and proved. In Section 4, it is seen that, if the expectation of the underlying random variable is not finite, as is the case in Theorem 7, a version of Theorem 7 is still true. However, if said expectation does not exist, then the averages are unbounded with probability 1. The chapter is concluded with a brief discussion of the tail σ-field of a sequence of random variables and pertinent results, including the Kolmogorov Zero–One Law for independent random variables, and the so-called Three Series Criterion.
Chapter 15 is not an entirely integral part of the body of basic and fundamental results of measure-theoretic probability. Rather, it is one of the many possible choices of topics that this author could have covered. It serves as a very brief introduction to an important class of discrete parameter stochastic processes—stationary and ergodic or nonergodic processes—with a view toward proving the fundamental result, the Ergodic Theorem. In this framework, the concept of a stationary stochastic process is introduced, and some characterizations of stationarity are presented. The convenient and useful coordinate process is also introduced at this point. Next, the concepts of a transformation as well as a measure-preserving transformation are discussed, and it is shown that a measure-preserving transformation along with an arbitrary random variable define a stationary process. Associated with each transformation is a class of invariant sets and a class of almost sure invariant sets, both of which are σ-fields. A special class of transformations is the class of ergodic transformations, which are defined at this point. Invariance with respect to a transformation can also be defined for a stationary sequence of random variables, and it is so done. At this point, all the required machinery is available for the formulation of the Ergodic Theorem; also, its proof is presented, after some additional preliminary results are established. In the final section of the chapter, invariance of sets and of random variables is defined relative to a stationary process. Also, an alternative form of the Ergodic Theorem is given for nonergodic as well as ergodic processes. In closing, it is to be pointed out that one direction of the Kolmogorov Strong Law of Large Numbers is a special case of the Ergodic Theorem, as a sequence of independent identically distributed random variables forms a stationary and ergodic process.
Throughout the years, this author has drawn upon a number of sources in organizing his lectures. Some of those sources are among the books listed in the Selected References Section. However, the style and spirit of the discussions in this book lie closest to those of Loève’s book. At this point, I would like to mention a recent worthy addition to the literature in measure theory—the book by Eric Vestrup, not least because Eric was one of our Ph.D. students at the University of California, Davis.
The lecture notes that eventually resulted in this book were revised, modified, and supplemented several times throughout the years; comments made by several of my students were very helpful in this respect. Unfortunately, they will have to remain anonymous, as I have not kept a complete record of them, and I do not want to provide an incomplete list. However, I do wish to thank my colleague and friend Costas Drossos for supplying a substantial number of exercises, mostly accompanied by
answers. I would like to thank the following reviewers: Ibrahim Ahmad, University of Central Florida; Richard Johnson, University of Wisconsin; Madan Puri, Indiana University; Doraiswamy Ramachandran, California State University at Sacramento; and Zongwu Cai, University of North Carolina at Charlotte. Finally, thanks are due to my Project Assistant Newton Wai, who very skillfully turned my manuscript into an excellent typed text.
George G. Roussas Davis, California November 2003
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CertainClassesofSets, Measurability,andPointwise
Approximation
Inthisintroductorychapter,theconceptsofafieldandofa σ -fieldareintroduced,they areillustratedbymeansofexamples,andsomerelevantbasicresultsarederived.Also, theconceptofamonotoneclassisdefinedanditsrelationshiptocertainfieldsand σ -fieldsisinvestigated.Givenacollectionofmeasurablespaces,theirproductspace isdefined,andsomebasicpropertiesareestablished.Theconceptofameasurable mappingisintroduced,anditsrelationtocertain σ -fieldsisstudied.Finally,itis shownthatanyrandomvariableisthepointwiselimitofasequenceofsimplerandom variables.
1.1 MeasurableSpaces
Let beanabstractset(orspace)andlet C beaclassofsubsetsof ;i.e., C ⊆ P ( ), theclassofallsubsetsof
Definition1. C issaidtobea field,usuallydenotedby F ,if (i) C isnonempty.
(ii) If A ∈ C ,then A c ∈ C .
(iii) If A 1 , A 2 ∈ C ,then A 1 ∪ A 2 ∈ C .
Remark1. Inviewof(ii)and(iii),theunion A 1 ∪ A 2 maybereplacedbythe intersection A 1 ∩ A 2
Examples. (Recallthatasetis countable ifitiseitherfiniteorithasthesame cardinalityasthesetofintegers.Inthelattercaseitis countablyinfinite.Asetis uncountable ifithasthesamecardinalityastherealnumbers.)
(1) C ={ , } isafieldcalledthe trivial field.(Itisthesmallestpossiblefield.)
(2) C ={allsubsetsof }= P ( ) isafieldcalledthe discrete field.(Itisthelargest possiblefield.)
(3) C ={ , A , A c , } forsome A with ⊂ A ⊂ .
(4) Let beinfinite(countablyornot)andlet C ={ A ⊆ ; A isfiniteor A c is finite}.Then C isafield.
(5) Let C betheclassofall(finite)sums(unionsofpairwisedisjointsets)ofthe partitioningsetsofafinitepartitionofanarbitraryset (see Definition12 below).Then C isafield(induced or generated bytheunderlyingpartition).
AnIntroductiontoMeasure-TheoreticProbability,SecondEdition. http://dx.doi.org/10.1016/B978-0-12-800042-7.00001-3
Copyright©2014ElsevierInc.Allrightsreserved.
Remark2. In Example4,itistobeobservedthatif isfiniteratherthaninfinite, then C = P ( ). ConsequencesofDefinition1.
(1) , ∈ F forevery F .
(2) If A j ∈ F , j = 1,..., n ,then n j =1 A j ∈ F .
(3) If A j ∈ F , j = 1,..., n ,then n j =1 A j ∈ F .
Remark3. Itisshownbyexamplesthat A j ∈ F , j ≥ 1,neednotimply ∞ j =1 A j ∈ F ,andsimilarlyfor ∞ j =1 A j (see Remark5 below).
Definition2. C issaidtobea σ -field ,usuallydenotedby A,ifitisafieldand(iii) in Definition1 isstrengthenedto
(iii ) If A j ∈ C , j = 1, 2,...,then ∞ j =1 A j ∈ C
Remark4. Inviewof(ii)and(iii),theunionin(iii )maybereplacedbythe intersection ∞ j =1 A j . Examples.
(6) C ={ , } isa σ -fieldcalledthe trivial σ -field. (7) C = P ( ) isa σ -fieldcalledthe discrete σ -field.
(8) Let beuncountableandlet C ={ A ⊆ ; A iscountableor A c iscountable}. Then C isa σ -field.(Ofcourse,if iscountablyinfinite,then C = P ( )).
(9) Let C betheclassofallcountablesumsofthepartitioningsetsofacountable partitionofanarbitraryset .Then C isa σ -field(induced or generated bythe underlyingpartition).
Remark5. A σ -fieldisalwaysafield,butafieldneednotbea σ -field.Infact, in Example4 take = (realline),andlet A j ={k integer; j ≤ k ≤ j }, j = 0, 1,... Then A j ∈ C , n j =0 A j ∈ C forany n = 0, 1,... but ∞ j =0 A j (= set ofallintegers) / ∈ C .
Let I beanyindexset.Then
Theorem1.
(i) If F j , j ∈ Iarefields,sois j ∈ I F j ={ A ⊆ ; A ∈ F j , j ∈ I }.
(ii) If A j , j ∈ Iare σ -fields,sois j ∈ I A j ={ A ⊆ ; A ∈ A j , j ∈ I }
Proof. Immediate.
Let C beanyclassofsubsetsof .Then Theorem2.
(i) Thereisauniqueminimalfieldcontaining C .Thisisdenotedby F (C ) andis calledthe fieldgeneratedby C
(ii) Thereisauniqueminimal σ -fieldcontaining C .Thisisdenotedby σ(C ) andis calledthe σ -fieldgeneratedby C .
Proof.
(i) F (C ) = j ∈ I F j ,where {F j , j ∈ I } isthenonemptyclassofallfieldscontaining C . (ii) σ(C ) = j ∈ I A j ,where {A j , j ∈ I } isthenonemptyclassofall σ -fields containing C .
Remark6. Clearly, σ(F (C )) = σ(C ).Indeed, C ⊆ F (C ),whichimplies σ(C ) ⊆ σ(F (C )).Also,forevery σ -field Ai ⊇ C itholds Ai ⊇ F (C ),since Ai isafield(being a σ -field),and F (C ) istheminimalfield(over C ).Hence σ(C ) = i Ai ⊇ F (C ) Since σ(C ) isa σ -field,itcontainstheminimal σ -fieldover F (C ),σ(F (C ));i.e., σ(C ) ⊇ σ(F (C )).Hence σ(C ) = σ(F (C ))
Application1. Let = and C0 ={allintervalsin }={( x , y ),( x , y ], [ x , y ), [ x , y ],(−∞, a ),(−∞, a ],(b , ∞), [b , ∞); x , y ∈ , x < y , a , b ∈ }.Then σ(C0 ) isdenotedby B andiscalledthe Borel σ -field overtherealline.Thesetsin B are called Borelsets.Let = ∪{−∞, ∞}. iscalledthe extended reallineandthe σ -field B generatedby B ∪{−∞}∪{∞} the extended Borel σ -field.
Remark7. { x }∈ B forevery x ∈ .Indeed, { x }= ∞ n =1 x , x + 1 n with x , x + 1 n ∈ B .Hence ∞ n =1 x , x + 1 n ∈ B ,or { x }∈ B .Alternatively,with a < x < b ,wehave { x }= (a , x ]∩[ x , b ) ∈ B
Definition3. Thepair ( , A) iscalleda measurablespace andthesetsin A measurablesets.Inparticular, ( , B ) iscalledthe Borelrealline,and ( , B ) the extendedBorelrealline.
Let C againbeaclassofsubsetsof .Then
Definition4. C iscalleda monotone classif A j ∈ C , j = 1, 2,... and A j ↑ (i.e., A 1 ⊆ A 2 ⊆··· ) or A j ↓ (i e., A 1 ⊇ A 2 ⊇··· ),thenlim j →∞ A j def = ∞ j =1 A j ∈ C andlim j →∞ A j def = ∞ j =1 A j ∈ C ,respectively.
Theorem3. A σ -field A isamonotonefield(i.e.,afieldthatisalsoamonotone class)andconversely.
Proof. Onedirectionisimmediate.Asfortheother,let F beamonotonefieldand letany A j ∈ F , j = 1, 2,....Toshowthat ∞ j =1 A j ∈ F .Wehave: ∞ j =1 A j =
A 1 ∪ ( A 1 ∪ A 2 ) ∪···∪ ( A 1 ∪···∪ A n ) ∪···= ∞ n =1 Bn ,where Bn = n j =1 A j , andhence Bn ∈ F , n = 1, 2,... and Bn ↑.Thus ∞ n =1 Bn ∈ F .
Theorem4. If M j , j ∈ I ,aremonotoneclasses,sois j ∈ I M j ={ A ⊆ ; A ∈ M j , j ∈ I }.
Proof. Immediate.
Theorem5. Thereisauniqueminimalmonotoneclass M containing C .
Proof. M = j ∈ I M j ,where {M j , j ∈ I } isthenonemptyclassofallmonotone classescontaining C .
Remark8. {M j , j ∈ I } isnonemptysince σ(C ) or P ( ) belonginit.
Remark9. Itmaybeseenbymeansofexamples(see Exercise12)thatamonotone classneednotbeafield.
However,seethenextlemma,aswellas Theorem6.
Lemma1. Let C beafieldand M betheminimalmonotoneclasscontaining C Then M isafield.
Proof. Inordertoprovethat M isafield,itsufficestoprovethatrelations(*)hold, where
(∗) ⎧ ⎨
forevery A , B ∈ M, wehave:
(i) A ∩ B ∈ M
(ii) A c ∩ B ∈ M
(iii) A ∩ B c ∈ M
(Thatis,forevery A , B ∈ M,theirintersectionisin M,andsoistheintersection ofanyoneofthembythecomplementoftheother.)
Infact, M ⊇ C ,implies ∈ M.Taking B = ,wegetthatforevery A ∈ M, A c ∩ = A c ∈ M (by(ii)).Sincealso A ∩ B ∈ M (by(i))forall A , B ∈ M,the proofwouldbecompleted.
Inordertoestablish(*),wefollowthefollowingthreesteps:
Step1. Forany A ∈ M,define M A ={ B ∈ M;(*)holds},sothat M A ⊆ M Obviously A ∈ M A ,since ∈ M.Itisassertedthat M A isa monotoneclass.Let B j ∈ M A , j = 1, 2,..., B j ↑.Toshowthat ∞ j =1 B j def = B ∈ M A ;i.e.,toshow that(*)holds.Wehave: A ∩ B = A ∩ ( j B j ) = j ( A ∩ B j ) ∈ M,since M is monotoneand A ∩ B j ↑.Next, A c ∩ B = A c ∩ ( j B j ) = j ( A c ∩ B j ) ∈ M since A c ∩ B j ∈ M,by(*)(ii),and A c ∩ B j ↑.Finally, A ∩ B c = A ∩ ( j B j )c = A ∩ ( j B c j ) = j ( A ∩ B c j ) with A ∩ B c j ∈ M by(*)(iii)and A ∩ B c j ↓,sothat j ( A ∩ B c j ) ∈ M since M ismonotone.Thecasethat B j ↓ istreatedsimilarly,and theproofthat M A isamonotoneclassiscomplete.
Step2.IfA ∈ C , then M A = M.Asalreadymentioned, M A ⊆ M.Soitsuffices toprovethat M ⊆ M A .Let B ∈ C .Then(*)holdsandhence B ∈ M A .Therefore C ⊆ M A .Bystep1, M A isamonotoneclassand M istheminimalmonotoneclass containing C .Thus M ⊆ M A andhence M A = M.
Step3. IfAisanysetin M, then M A = M.Weshowthat C ⊆ M A ,which implies M ⊆ M A since M A isamonotoneclasscontaining C and M istheminimal monotoneclassover C .Sincealso M A ⊆ M,theresult M A = M wouldfollow. Toshow C ⊆ M A ,take B ∈ C andconsider M B .Then M B = M bystep2.Since A ∈ M,wehave A ∈ M B ,whichimpliesthat B ∩ A , B c ∩ A ,and B ∩ A c allbelong in M;or A ∩ B , A c ∩ B ,and A ∩ B c belongin M,whichmeansthat B ∈ M A .
Theorem6. Let C beafieldand M betheminimalmonotoneclasscontaining C . Then M = σ(C ).
Proof. Evidently, M ⊆ σ(C ) sinceevery σ -fieldisamonotoneclass.By Lemma1, M isafield,andhencea σ -field,by Theorem3.Thus, M ⊇ σ(C ) andhence M = σ(C ).
Remark10. Lemma1 and Theorem6 justdiscussedprovideanillustrationofthe intricaterelationoffields,monotoneclasses,and σ -fieldsinacertainsetting.Aswill alsobeseeninseveralplacesinthisbook,monotoneclassesareoftenusedastools inargumentsmeanttoestablishresultsabout σ -fields.Inthiskindofarguments, therolesofafieldandofamonotoneclassmaybesubstitutedbytheso-called π -systems and λ-systems,respectively.Thedefinitionoftheseconceptsmaybefound, forexample,inpage41in Billingsley(1995).Aresultanalogousto Theorem6 is thenTheorem1.3inpage5ofthereferencejustcited,whichstatesthat:If P isa π -systemand G isa λ-system,then P ⊂ G implies σ(P ) ⊂ G
1.2 ProductMeasurableSpaces
Considerthemeasurablespaces ( 1 , A1 ),( 2 , A2 ).Then
Definition5. The productspace of 1 , 2 ,denotedby 1 × 2 ,isdefined asfollows: 1 × 2 ={ω = (ω1 ,ω2 ); ω1 ∈ 1 ,ω2 ∈ 2 }.Inparticular,for A ∈ A1 , B ∈ A2 theproductof A , B ,denotedby A × B ,isdefinedby: A × B ={ω = (ω1 ,ω2 ); ω1 ∈ A ,ω2 ∈ B },andthesubsets A × B of 1 × 2 for A ∈ A1 , B ∈ A2 arecalled(measurable) rectangles. A , B arecalledthe sides oftherectangle.
From Definition5,oneeasilyverifiesthefollowinglemma.
Lemma2. Considertherectangle E = A × B .Then,with“+”denotingunionof disjointevents,
(i) E c = ( A × B c ) + ( A c × 2 ) = ( A c × B ) + ( 1 × B c ).
Considertherectangles E 1 = A 1 × B1 , E 2 = A 2 × B2 .Then
(ii) E 1 ∩ E 2 = ( A 1 ∩ A 2 ) × ( B1 ∩ B2 ).Hence E 1 ∩ E 2 = ifandonlyifatleast oneofthesets A 1 ∩ A 2 , B1 ∩ B2 is .
Considertherectangles E 1 , E 2 asabove,andtherectangles F1 = A 1 × B1 , F2 = A 2 × B2 .Then (iii) ( E 1 ∩ F1 ) ∩ ( E 2 ∩ F2 ) =[( A 1 ∩ A 1 ) × ( B1 ∩ B1 )]∩[( A 2 ∩ A 2 ) ×( B2 ∩ B2 )] (by(ii)) =[( A 1 ∩ A 1 ) ∩ ( A 2 ∩ A 2 )] ×[( B1 ∩ B1 ) ∩ ( B2 ∩ B2 )] (by(ii)) =[( A 1 ∩ A 2 ) ∩ ( A 1 ∩ A 2 )] ×[( B1 ∩ B2 ) ∩ ( B1 ∩ B2 )]
Hence,theleft-handsideis ifandonlyifatleastoneof ( A 1 ∩ A 2 ) ∩ ( A 1 ∩ A 2 ),( B1 ∩ B2 ) ∩ ( B1 ∩ B2 ) is Theorem7. Let C betheclassofallfinite sums (i.e.,unionsofpairwisedisjoint)ofrectangles A × B with A ∈ A1 , B ∈ A2 .Then C isafield(ofsubsetsof 1 × 2 ).
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places by a layer of mutton-fat, half an inch thick, which filled up the intervening spaces. His hollow cheeks, deep-sunken eyes, thin and wiry beard, and the long spear he carried in his hand made him a fair representative of Don Quixote, and the resemblance was not diminished by the gaunt and ungainly camel on which he jogged along at the head of my caravan. He was very devout, praying for quite an unreasonable length of time before and after meals, and always had a large patch of sand on his forehead, from striking it on the ground, as he knelt towards Mecca. Both his arms, above the elbows, were covered with rings of hippopotamus hide, to which were attached square leathern cases, containing sentences of the Koran, as charms to keep away sickness and evil spirits. The other man, Saïd, was a Shygheean, willing and good-natured enough, but slow and regardless of truth, as all Arabs are. Indeed, the best definition of an Arab which I can give, is—a philosophizing sinner. His fatalism gives him a calm and equable temperament under all circumstances, and “God wills it!” or “God is merciful!” is the solace for every misfortune. But this same carelessness to the usual accidents of life extends also to his speech and his dealings with other men. I will not say that an Arab never speaks truth: on the contrary, he always does, if he happens to remember it, and there is no object to be gained by suppressing it; but rather than trouble himself to answer correctly a question which requires some thought, he tells you whatever comes uppermost in his mind, though certain to be detected the next minute. He is like a salesman, who, if he does not happen to have the article you want, offers you something else, rather than let you go away empty-handed. In regard to his dealings, what Sir Gardner Wilkinson says of Egypt, that “nobody parts with money without an effort to defraud,” is equally true of Nubia and Soudân. The people do not steal outright; but they have a thousand ways of doing it in an indirect and civilized manner, and they are perfect masters of all those petty arts of fraud which thrive so greenly in the great commercial cities of Christendom. With these slight drawbacks, there is much to like in the Arabs, and they are certainly the most patient, assiduous and good-humored people in the world. If they fail in cheating you, they respect you the more,
and they are so attentive to you, so ready to take their mood from yours—to laugh when you are cheerful, and be silent when you are grave—so light-hearted in the performance of severe duties, that if you commence your acquaintance by despising, you finish by cordially liking them.
On a journey like that which I was then commencing, it is absolutely necessary to preserve a good understanding with your men and beasts; otherwise travel will be a task, and a severe one, instead of a recreation. After my men had vainly tried a number of expedients, to get the upper hand of me, I drilled them into absolute obedience, and found their character much improved thereby. With my dromedary, whom I called Abou-Sin, (the Father of Teeth), from the great shekh of the Shukoree Arabs, to whom he originally belonged, I was soon on good terms. He was a beast of excellent temper, with a spice of humor in his composition, and a fondness for playing practical jokes. But as I always paid them back, neither party could complain, though Abou-Sin sometimes gurgled out of his long throat a string of Arabic gutturals, in remonstrance. He came up to my tent and knelt at precisely the same hour every evening, to get his feed of dourra, and when I was at breakfast always held his lips pursed up, ready to take the pieces of bread I gave him. My men, whom I agreed to provide with food during the journey, were regaled every day with mutton and mareesa, the two only really good things to be found in Soudân. A fat sheep cost 8 piastres (40 cents), and we killed one every three days. The meat was of excellent flavor. Mareesa is made of the coarse grain called dourra, which is pounded into flour by hand, mixed with water, and heated over a fire in order to produce speedy fermentation. It is always drunk the day after being made, as it turns sour on the third day. It is a little stronger than small beer, and has a taste similar to wheat bran, unpleasant on the first trial and highly palatable on the second. A jar holding two gallons costs one piastre, and as few families, however poor, are without it, we always found plenty of it for sale in the villages. It is nutritious, promotive of digestion, and my experience went to prove that it was not only a harmless but
most wholesome drink in that stifling climate. Ombilbil, the mother of nightingales, which is made from wheat, is stronger, and has a pungent flavor. The people in general are remarkably temperate, but sailors and camel-men are often not content without arakee, a sort of weak brandy made from dates. I have heard this song sung so often that I cannot choose but recollect the words. It is in the Arabic jargon of Soudân:
“El-toombak sheràboo dowaïa, Oo el karafeen ed dowa il ’es-sufaïa, Oo el àrakee legheetoo monnaïa, Om bilbil bukkoosoo burraïa.”
[Tobacco I smoke in the pipe; and mareesa is a medicine to the sufaïa; (i.e.the bag of palm fibres through which it is strained), but arakee makes me perfectly contented, and then I will not even look at bilbil].
The third day after leaving Khartoum, I reached the mountains of Gerri, through which the Nile breaks his way in a narrow pass. Here I hailed as an old acquaintance the island-hill of Rowyàn (the watered, or unthirsty). This is truly a magnificent peak, notwithstanding its height is not more than seven hundred feet. Neither is Soracte high, yet it produces a striking effect, even with the loftier Apennines behind it. The Rowyàn is somewhat similar to Soracte in form. There are a few trees on the top, which shows that there must be a deposit of soil above its barren ramparts, and were I a merchant of Khartoum I should build a summer residence there, and by means of hydraulics create a grove and garden around it. The akaba, or desert pass, which we were obliged to take in order to reach the river again, is six hours in length, through a wild, stony tract, covered with immense boulders of granite, hurled and heaped together in the same chaotic manner as is exhibited in the rocks between Assouan and Philæ. After passing the range, a wide plain again opened before us, the course of the Nile marked in its centre by the darker hue of the nebbuks and sycamores, rising above the
long gray belts of thorn-trees. The mountains which inclose the fallen temples of Mesowuràt and Naga appeared far to the east. The banks of the river here are better cultivated than further up the stream. The wheat, which was just sprouting, during my upward journey, was now two feet high, and rolled before the wind in waves of dark, intense, burning green. The brilliancy of color in these midAfrican landscapes is truly astonishing.
The north-wind, which blew the sand furiously in our faces during the first three days of the journey, ceased at this point and the weather became once more intensely hot. The first two or three hours of the morning were, nevertheless, delicious. The temperature was mild, and there was a June-like breeze which bore far and wide the delicate odor of the mimosa blossoms. The trees were large and thick, as on the White Nile, forming long, orchard-like belts between the grain-fields and the thorny clumps of the Desert. The flocks of black goats which the natives breed, were scattered among these trees, and numbers of the animals stood perfectly upright on their hind legs, as they nibbled off the ends of the higher branches.
On the morning after leaving Akaba Gerri, I had two altercations with my men. Mohammed had left Khartoum without a camel, evidently for the purpose of saving money. In a day or two, however, he limped so much that I put him upon Achmet’s dromedary for a few hours. This was an imposition, for every guide is obliged to furnish his own camel, and I told the old man that he should ride no more. He thereupon prevailed upon Saïd to declare that their contract was to take me to Ambukol, instead of Merawe. This, considering that the route had been distinctly stated to them by Dr. Reitz, in my presence, and put in writing by the moodir, Abdallah Effendi, and that the name of Ambukol was not once mentioned, was a falsehood of the most brazen character. I told the men they were liars, and that sooner than yield to them I would return to Khartoum and have them punished, whereupon they saw they had gone too far, and made a seeming compromise by declaring that they would willingly take me to Merawe, if I wished it.
Towards noon we reached the village of Derreira, nearly opposite the picturesque rapids of the Nile. I gave Mohammed half a piastre and sent him after mareesa, two gallons of which he speedily procured. A large gourd was filled for me, and I drank about a quart without taking breath. Before it had left my lips, I experienced a feeling of vigor and elasticity throughout my whole frame, which refreshed me for the remainder of the day. Mohammed stated that the tents of some of his tribe were only about four hours distant, and asked leave to go and procure a camel, promising to rejoin us at El Metemma the next day. As Saïd knew the way, and could have piloted me in case the old sinner should not return, I gave him leave to go.
Achmet and I rode for nearly two hours over a stony, thorny plain, before we overtook the baggage camels. When at last we came in sight of them, the brown camel was running loose without his load and Saïd trying to catch him. My provision-chests were tumbled upon the ground, the cafass broken to pieces and the chickens enjoying the liberty of the Desert. Saïd, it seemed, had stopped to talk with some women, leaving the camel, which was none too gentle, to take care of himself. Achmet was so incensed that he struck the culprit in the face, whereupon he cried out, with a rueful voice: “ya khosara!” (oh, what a misfortune!). After half an hour’s labor the boxes were repacked, minus their broken crockery, the chickens caught and the camel loaded. The inhabitants of this region were mostly Shygheeans, who had emigrated thither. They are smaller and darker than the people of Màhass, but resemble them in character. In one of the villages which we passed, the soog, or market, was being held. I rode through the crowd to see what they had to sell, but found only the simplest articles: camels, donkeys, sheep, goats; mats, onions, butter, with some baskets of raw cotton and pieces of stuff spun and woven by the natives. The sales must be principally by barter, as there is little money in the country.
In the afternoon we passed another akaba, even more difficult for camels than that of Gerri. The tracks were rough and stony, crossed by frequent strata of granite and porphyry. From the top of one of
the ridges I had a fine view of a little valley of mimosas which lay embayed in the hills and washed by the Nile, which here curved grandly round from west to south, his current glittering blue and broad in the sun. The opposite bank was flat and belted with wheat fields, beyond which stretched a gray forest of thorns and then the yellow savannas of Shendy, walled in the distance by long, blue, broken ranges of mountains. The summit of a hill near our road was surrounded with a thick wall, formed of natural blocks of black porphyry. It had square, projecting bastions at regular intervals, and an entrance on the western side. From its appearance, form and position, it had undoubtedly been a stronghold of some one of the Arab tribes, and can claim no great antiquity. I travelled on until after sunset, when, as no village appeared, I camped in a grove of large mimosas, not far from the Nile. A few Shygheean herdsmen were living in brush huts near at hand, and dogs and jackals howled incessantly through the night.
On the fifth day I reached the large town of El Metemma, nearly opposite Shendy, and the capital of a negro kingdom, before the Egyptian usurpation. The road, on approaching it, leads over a narrow plain, covered with a shrub resembling heather, bordered on one side by the river, and on the other by a long range of bare red sand-hills. We journeyed for more than three hours, passing point after point of the hills, only to find other spurs stretching out ahead of us. From the intense heat I was very anxious to reach El Metemma, and was not a little rejoiced when I discerned a grove of date-trees, which had been pointed out to me from Shendy, a month before, as the landmark of the place. Soon a cluster of buildings appeared on the sandy slopes, but as we approached, I saw they were ruins. We turned another point, and reached another group of tokuls and clay houses—ruins also. Another point, and more ruins, and so for more than a mile before we reached the town, which commences at the last spur of the hills, and extends along the plain for a mile and a half.
It is a long mass of one-story mud buildings, and the most miserable place of its size that I have seen in Central Africa. There is
no bazaar, but an open market-place, where the people sit on the ground and sell their produce, consisting of dourra, butter, dates, onions, tobacco and a few grass mats. There may be a mosque in the place, but in the course of my ramble through the streets, I saw nothing that looked like one. Half the houses appeared to be uninhabited, and the natives were a hideous mixture of the red tribes of Màhass and Shygheea and the negro races of Soudân. A few people were moving lazily through the dusty and filthy lanes, but the greater portion were sitting in the earth, on the shady side of the houses. In one of the streets I was taken for the Medical Inspector of the town, a part of whose business it is to see that it is kept free from filth. Two women came hastily out of the houses and began sweeping vigorously, saying to me as I came up: “You see, we are sweeping very clean.” It would have been much more agreeable to me, had the true Inspector gone his rounds the day before. El Metemma and Shendy are probably the most immoral towns in all Central Africa. The people informed me that it was a regular business for persons to buy female slaves, and hire them for the purpose of prostitution, all the money received in this vile way going into the owner’s pocket.
I was occupied the rest of the day and the next morning in procuring and filling additional water-skins, and preparing to cross the Beyooda. Achmet had a quantity of bread baked, for the journey would occupy seven or eight days, and there was no possibility of procuring provisions on the road. Mohammed did not make his appearance at the appointed time, and I determined to start without him, my caravan being increased by a Dongolese merchant, and a poor Shygheean, whose only property was a club and a wooden bowl, and who asked leave to help tend the camels for the sake of food and water on the way. All of the Beyooda, which term is applied to the broad desert region west of the Nile and extending southward from Nubia to Kordofan and Dar-Fūr, is infested with marauding tribes of Arabs, and though at present their depredations are less frequent than formerly, still, from the total absence of all protection, the traveller is exposed to considerable risk. For this reason, it is not
usual to find small parties traversing this route, as in the Nubian Desert.
I added to my supplies a fat sheep, a water-skin filled with mareesa, a sheaf of raw onions (which are a great luxury in the Desert), and as many fowls as could be procured in El Metemma. Just as we were loading the camels, who should come up but Beshir and two or three more of the Mahassee sailors, who had formed part of my crew from Berber to Khartoum. They came up and kissed my hand, exclaiming: “May God prosper you, O Effendi!” They immediately set about helping to load the camels, giving us, meanwhile, news of every thing that had happened. Beshir’s countenance fell when I asked him about his Metemma sweetheart, Gammerò-Betahadjerò; she had proved faithless to him. The America was again on her way from Berber to Khartoum, with a company of merchants. The old slave, Bakhita, unable to bear the imputation of being a hundred and fifty years old, had run away from the vessel. When the camels were loaded and we were ready to mount, I gave the sailors a few piastres to buy mareesa and sent them away rejoicing.
