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Appendix A. Measure and Integration

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Appendix A. Measure and Integration Appendix A. Measure and integration We suppose the reader is familiar with the basic facts concerning set theory and integration as they are presented in the introductory course of analysis. In this appendix, we review them briefly, and add some more which we shall need in the text. Basic references for proofs and a detailed exposition are, e.g., [[ H a l 1 ]] , [[ J a r 1 , 2 ]] , [[ K F 1 , 2 ]] , [[ L i L ]] , [[ R u 1 ]] , or any other textbook on analysis you might prefer. A.1 Sets, mappings, relations A set is a collection of objects called elements. The symbol card X denotes the cardi- nality of the set X. The subset M consisting of the elements of X which satisfy the conditions P1(x),...,Pn(x) is usually written as M = { x ∈ X : P1(x),...,Pn(x) }.A set whose elements are certain sets is called a system or family of these sets; the family of all subsystems of a given X is denoted as 2X . The operations of union, intersection, and set difference are introduced in the standard way; the first two of these are commutative, associative, and mutually distributive. In a { } system Mα of any cardinality, the de Morgan relations , X \ Mα = (X \ Mα)and X \ Mα = (X \ Mα), α α α α are valid. Another elementary property is the following: for any family {Mn} ,whichis { } at most countable, there is a disjoint family Nn of the same cardinality such that ⊂ \ ∪ \ Nn Mn and n Nn = n Mn.Theset(M N) (N M) is called the symmetric difference of the sets M,N and denoted as M #N. It is commutative, M #N = N #M, and furthermore, we have M #N =(M ∪N)\(M ∩N)andM #N =(X\M)#(X\N)for any X ⊃ M∪N. The symmetric difference is also associative, M#(N#P )=(M#N)#P , and distributive with respect to the intersection, (M # N) ∩ P =(M ∩ P ) # (N ∩ P ). A family R is called a set ring if M # N ∈R and M ∩ N ∈R holds for any pair M,N ∈R. The relation M \ N =(M # N) ∩ M also gives M \ N ∈R, and this in turn implies ∅∈R and M ∪ N ∈R. If the symmetric difference and intersection are understood as a sum and product, respectively, then a set ring is a ring in the sense of the general algebraic definition of Appendix B.1. A.1.1 Example: Let J d be the family of all bounded intervals in Rd,d≥ 1. The family Rd, which consists of all finite unions of intervals J ⊂Jd together with the empty set, is a set ring, and moreover, it is the smallest set ring containing J d. As mentioned above, any R ∈Rd can be expressed as a finite union of disjoint bounded intervals. 595 596 Appendix A Measure and integration A set ring R⊂2X is called a set field if it contains the set X (notice that in the terminology of Appendix B.1 it is a ring with a unit element but not an algebra). A set field A⊂ X ∞ ∈A { }∞ ⊂A 2 is called a σ–field if n=1 Mn holds for any countable system Mn n=1 . ∞ ∈A A⊂ X De Morgan relations show that also n=1 Mn , and furthermore that a family 2 \ ∈A ∈A ∈A containing the set X is a σ–field iff X M for all M and n Mn for any X at most countable subsystem {Mn}⊂A. Given a family S⊂2 we consider all σ–fields A⊂2X containing S (there is at least one, A =2X ). Their intersection is again a σ–field containing S;wecallittheσ–field generated by S and denote it as A(S). A.1.2 Example: The elements of Bd := A(J d) are called Borel sets in Rd. In particular, all the open and closed sets, and thus also the compact sets, are Borel. The σ–field Bd is also generated by other systems, e.g., by the system of all open sets in Rd. In general, Borel sets in a topological space (X, τ) are defined as the elements of the σ–field A(τ). { }∞ ⊂ ⊃ A sequence Mn n=1 is nondecreasing or nonincreasing if Mn Mn+1 or Mn M Mn+1, respectively, holds for n =1, 2,.... A set family is monotonic if it contains { } the set n Mn together with any nondecreasing sequence Mn ,and n Mn together with any nonincreasing sequence {Mn}.Anyσ–field represents an example of a monotonic system. To any S there is the smallest monotonic system M(S) containingS and we M S ⊂AS R M R ∈ have ( ) ( ). If is a ring, the same is true for ( ); in addition, if M∈R M M(R), then M(R)isaσ–field and M(R)=A(R). A mapping (or map) f from a set X to Y is a rule, which associates with any x ∈ X a unique element y ≡ f(x) of the set Y ;wewrite f : X → Y and also x → f(x). If Y = R or Y = C the map f is usually called a real or a complex function, respectively. It is also often useful to consider maps which are defined on a subset Df ⊂ X only. The symbol f : X → Y must then be completed by specifying the set Df which is called the domain of f;wedenoteitalsoas D(f). If Df is not specified, it is supposed to coincide with X. The sets Ran f := { y ∈ Y : y = f(x),x∈ Df } and Ker f := { x ∈ Df : f(x)=0} are the range and kernel of the map f, respectively. A map f : X → Y is injective if f(x)=f(x) holds for any x, x ∈ X only if x = x; it is surjective if Ran f = Y . A map which is simultaneously injective and surjective is called bijective or a bijection. The sets X and Y have the same cardinality if there is a bijection f : X → Y with Df = X. The relation f = g between f : X → Y and g : X → Y means by definition Df = Dg and f(x)=g(x) for all x ∈ Df .If Df ⊃ Dg and f(x)=g(x) holds for all x ∈ Dg we say that f is an extension of g while g is a \ restriction of f to the set Dg;wewrite f ⊃ g and g = f | Dg. A.1.3 Example: For any X ⊂ M we define a real function χM : χM (x)=1ifx ∈ M, χM (x)=0ifx ∈ X \ M; it is called the characteristic (or indicator) function of the X set M. The map M → χM is a bijection of the system 2 to the set of all functions f : X → R such that Ran f = {0, 1}. (−1) { ∈ ∈ } → ⊂ The set f (N):= x Df : f(x) N for givenf : X Y and N Y is called (−1) (−1) the pull–back of the set N by the map f. One has f α∈I Nα = α∈I f (Nα)for any family {Nα}⊂Y , and the analogous relation is valid for intersections. Furthermore, (−1) (−1) (−1) (−1) f (N1 \ N2)=f (N1) \ f (N2)andf(f (N)=Ranf ∩ N. On the other hand, f (−1)(f(M)) ⊃ M; the inclusion turns to identity if f is injective. Y A.1.4 Example: Let f : X → Y with Df = X.Toaσ–field B⊂2 we can construct the family f (−1) := { f (−1)(N): N ∈B}, which is obviously again a σ–field. Similarly, if A.1 Sets, mappings, relations 597 A⊂2X is a σ–field, then the same is true for { N ⊂ Y : f (−1)(N) ∈A}. Hence for any family S⊂2Y we can construct the σ–field f (−1)(A(S)) ⊂ 2X and the latter coincides with the σ–field generated by f (−1)(S), i.e., f (−1)(A(S)) = A(f (−1)(S)). Given f : X → Y and g : Y → Z we can define the composite map g◦f : X → Z (−1) (−1) with the domain D(g◦f):=f (Dg)=f (Dg ∩ Ran f)by(g◦f)(x):=g(f(x)). We have (g◦f)(P )(−1) = f (−1)(g(−1)(P )) for any P ⊂ Z. If f : X → Y is injective, then for any y ∈ Ran f there is just one xy ∈ Df such that y = f(xy); the prescription y → g(y):=xy defines a map g : Y → X,whichis −1 −1 −1 called the inverse of f and denoted as f .Wehave D(f )=Ranf, Ran f = Df −1 −1 and f (f(x)) = x, f(f (y)) = y for any x ∈ Df and y ∈ Ran f, respectively. These relations further imply f (−1)(N)=f −1(N) for any N ⊂ Ran f.Oftenwehaveapairof mappings f : X → Y and g : Y → X and we want to know whether f is invertible and f −1 = g; this is true if one of the following conditions is valid: (i) Dg =Ranf and g(f(x)) = x for all x ∈ Df (ii) Ran f ⊂ Dg, g(f(x)) = x for all x ∈ Df and Ran g ⊂ Df , f(g(y)) = y for all y ∈ Dg If f : X → Y is injective, then f −1 is also injective and (f −1)−1 = f.If g : Y → Z is also injective, then the composite map g◦f is invertible and (g◦f)−1 = f −1◦g−1. The Cartesian product M × N is the set of ordered pairs [x, y]withx ∈ M and y ∈ N; the Cartesian product of the families S and S is defined by S×S := { M × N : M ∈S,N ∈S }.
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