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A Review of Some Basic Concepts of Topology and Measure Theory

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A Review of Some Basic Concepts of Topology and Measure Theory APPENDIX 1 A Review of Some Basic Concepts of Topology and Measure Theory In this appendix, we summarize, mainly without proof, some standard results from topology and measure theory. The aims are to establish terminology and notation, to set out results needed at various stages in the text in some specific form for convenient reference, and to provide some brief perspectives on the development of the theory. For proofs and further details, the reader should refer, in particular, to Kingman and Taylor (1966, Chapters 1–6), whose development and terminology we have followed rather closely. A1.1. Set Theory A set A of a space X is a collection of elements or points of X .Whenx is an element of the set A,wewritex ∈ A (x belongs to or is included in A). The set of points of X not included in A is the complement of A, written Ac.If A, B aretwosetsofpointsfromX ,theirunion, written A ∪ B, is the set of points in either A or B or both; their symmetric difference, written A B, is the set of points in A or B but not both. If every element of B is also an element of A,wesayB is included in A (B ⊆ A)orA contains B (A ⊇ B). In this case, the proper difference of A and B, written either A − B or A \ B, is the set of points of A but not B. More generally, we use A − B for A ∩ Bc, so A − B = A B only when A ⊃ B. The operations ∩ and on subsets of X are commutative, associative and distributive. The class of all such subsets thus forms an algebra with respect to these operations, where ∅, the empty set, plays the role of identity for and X the role of identity for ∩. The special relation A ∩ A = A implies that the algebra is Boolean. More generally, any class of sets closed under the operations of ∩ and is called a ring,oranalgebra if X itself is a member of the class. A semiring is a class of sets A with the properties (i) A is closed under intersections and (ii) every symmetric difference of sets in A can be 368 A1.2. Topologies 369 represented as a finite union of disjoint sets in A. The ring generated by an arbitrary family of sets F is the smallest ring containing F or, equivalently, the intersection of all rings containing F. Every element in the ring generated by a semiring A can be represented as a union of disjoint sets of A.IfR is a finite ring, there exists a basis of disjoint elements of R such that every element in R can be represented uniquely as a union of disjoint elements of the basis. The notions of union and intersection can be extended to arbitrary classes { } ↑ of sets. If An: n =1, 2,... is a sequence of sets, write An A = lim An ⊆ ∞ ⊇ if An An+1 (n =1, 2,...)and A = n=1 An; similarly, if An An+1, ↓ ∞ write An A = lim An if A = n=1 An.Amonotone class is a class of sets closed under monotonically increasing sequences. A ring or algebra that is closed under countable unionsiscalledaσ-ring or σ-algebra, respectively. The σ-ring generated by a class of sets C, written σ(C), is the smallest σ- ring containing C.Aσ-ring is countably generated if it can be generated by a countable class of C. The following result, linking σ-rings to monotone classes, is useful in identifying the σ-ring generated by certain classes of sets. Proposition A1.1.I (Monotone Class Theorem). If R is a ring and C is a monotone class containing R,thenC contains σ(R). A closely related result uses the concept of a Dynkin system D meaning (i) X∈D; (ii) D is closed under proper differences; and (iii) D is closed under monotonically increasing limits. Proposition A1.1.II (Dynkin System Theorem). If S is a class of sets closed under finite intersections, and D is a Dynkin system containing S, then D contains σ(S). A1.2. Topologies A topology U on a space X is a class of subsets of X that is closed under arbitrary unions and finite intersections and that includes the empty set ∅ and the whole space X ;themembersofU are open sets, while their complements are closed sets. The pair (X , U)isatopological space.Theclosure of an arbitrary set A from X , written A¯, is the smallest closed set (equivalently, the intersection of all closed sets) containing A.Theinterior of A, written A◦, is the largest open set (equivalently, the union of all open sets) contained within A.Theboundary of A, written ∂A, is the difference A¯ \ A◦.The following elementary properties of boundaries are needed in the discussion of weak convergence of measures. Proposition A1.2.I. (a) ∂(A ∪ B) ⊆ ∂A ∪ ∂B; (b) ∂(A ∩ B) ⊆ ∂A ∪ ∂B; (c) ∂Ac = ∂A. 370 APPENDIX 1. Some Basic Topology and Measure Theory Concepts A neighbourhood of the point x ∈X with respect to the topology U (or, more briefly, a U-neighbourhood of x)isanopensetfromU containing x. U is a Hausdorff or T2-topology if the open sets separate points; that is, if for x = y, x and y possess disjoint neighbourhoods. A family of sets F forms a basis for the topology U if every U ∈U can be represented as a union of sets in F and F⊆U. U is then said to be generated by F. U is second countable if it has a countable basis. A sufficient condition for a family of sets to form a basis for some topology is that, if F1 ∈F, F2 ∈Fand x ∈ F1 ∩F2, then there exists F3 ∈Fsuch that x ∈ F3 ⊆ F1 ∩ F2. The topology generated by F is then uniquely defined and consists of all unions of sets in F. Two bases F and G,say,areequivalent if they generate the same topology. A necessary and sufficient condition for F and G to be equivalent is that for each F ∈F and x ∈ F , there exists G ∈G with x ∈ G ⊆ F , and similarly for each G ∈Gand y ∈ G, there exists F ∈F such that y ∈ F ⊆ G. Given a topology U on X , a notion of convergence of sequences (or more generally nets, but we do not need the latter concept) can be introduced by saying xn → x in the topology U if, given any U-neighbourhood of x, Ux,there exists an integer N (depending on the neighbourhood in general) such that xn ∈ Ux for n ≥ N. Conversely, nearly all the important types of convergence can be described in terms of a suitable topology. In this book, the overwhelm- ingemphasisisonmetric topologies, where the open sets are defined in terms of a metric or distance function ρ(·) that satisfies the conditions, for arbitrary x, y, z ∈X, (i) ρ(x, y)=ρ(y, x); (ii) ρ(x, y) ≥ 0andρ(x, y) = 0 if and only if x = y;and (iii) (triangle inequality) ρ(x, y)+ρ(y, z) ≥ ρ(x, z). With respect to a given distance function ρ,theopen sphere S (x) is the set {y: ρ(x, y) <}, being defined for any >0. For any set A, define its diameter by # $ diam A =2inf r: Sr(x) ⊇ A for some x . The metric topology generated by ρ is the smallest topology containing the open spheres; it is necessarily Hausdorff. A set is open in this topology if and only if every point in the set can be enclosed by an open sphere lying wholly within the set. A sequence of points {xn} converges to x in this topology if and only if ρ(xn,x) → 0. A limit point y of a set A is a limit of a sequence of points xn ∈ A with xn = y; y need not necessarily be in A. The closure of A in the metric topology is the union of A and its limit points. A space X with topology U is metrizable if a distance function ρ can be found such that U is equivalent to the metric topology generated by ρ. Two metrics on the same space X are equivalent if they each generate the same topology on X . A sequence of points {xn: n ≥ 1} in a metric space is a Cauchy sequence if ρ(xn,xm) → 0asn, m →∞. The space is complete if every Cauchy sequence has a limit; i.e. if for every Cauchy sequence {xn} there exists x ∈X such A1.2. Topologies 371 that ρ(xn,x) → 0. A set D is dense in X if, for every >0, every point in X can be approximated by points in D; i.e. given x ∈X, there exists d ∈Dsuch that ρ(x, d) <. The space X is separable if there exists a countable dense set, also called a separability set.IfX is a separable metric space, the spheres with rational radii and centres on a countable dense set form a countable base for the topology. Given two topological spaces (X1, U1)and(X2, U2), a mapping f(·)from −1 (X1, U1)to(X2, U2)iscontinuous if the inverse image f (U) of every open set U ∈U2 is an open set in U1. If both spaces are metric spaces, the mapping is continuous if and only if for every x ∈X1 and every >0, there exists δ>0 such that ρ2(f(x ),f(x)) <whenever ρ1(x ,x) <δ,whereρi is themetricinXi for i =1, 2; we can express this more loosely as f(x ) → f(x) whenever x → x.Ahomeomorphism is a one-to-one continuous-both- ways mapping between two topological spaces.
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