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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. Al.l. Set Theory A set A of a space !![ is a collection of elements or points of !!l. When x is an element of the set A, we write x E A (x belongs to or is included in A). The set of points of!![ not included in A is the complement of A, written A c. If A, B are two sets of points from !![ their union, written A u 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; and their intersection, written A n B, is the set of points in both. If every element of B is also an element of A we say B is included in A (B ~ A), or ·A contains B (A 2 B). In this case the proper difference of A and B, written either A - BorA \B, is the set of points of A but not B. More generally, we use A - B for A n Be, but A - B = A/::::,. B only when A2B. The operations n and /::::,. on subsets of!![ are commutative, associative, and distributive. The class of all such subsets thus forms an algebra with respect to these operations, where 0, the empty set, plays the role of identity for 6, Al.2. Topologies 593 and !!l the role of identity for n. The special relation AnA=A implies that the algebra is Boolean. More generally, any class of sets closed under the operations of n and !::,. is called a ring, or an algebra if !!l itself is a member of the class. A semiring is a class of sets .91 with the properties (i) .91 is closed under intersections, and (ii) every symmetric difference of sets in .91 can be represented as a finite union of disjoint sets in .91. The ring generated by an arbitrary family of sets 1F is the smallest ring containing 17, or, equiv­ alently, the intersection of all rings containing !F. Every element in the ring generated by a semiring .91 can be represented as a union of disjoint sets of .91. If qt is a finite ring, there exists a basis of disjoint elements of 9t such that every element in 9t 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, we write An j A = lim An if An£ An+l• n = 1, 2, ... ' and A= Uf An; similarly, if An 2 An+l• n = 1, 2, ... ' we write An! A = lim An if A = nf An. A monotone class is a class of sets closed under monotone increasing sequences. A ring (algebra) that is closed under countable as well as finite unions is called a a-ring (a-algebra). The a-ring generated by a class of sets rl, written a(rl), is the smallest a-ring containing rl. A a-ring is countably generated if it can be generated by a countable class ofrl. The following result, linking a-rings to monotone classes, is useful in identifying the a-ring generated by certain classes of sets. Proposition Al.l.I (Monotone Class Theorem). If 9t is a ring, and rl is a monotone class containing qt, then rl contains a(!Jt). A closely related result uses the concept of a Dynkin system: !?) is a Dynkin system if (i) !!£ E !?); (ii) !?) is closed under proper differences; and (iii) !?) is closed under monotone increasing limits. Proposition Al.l.II (Dynkin System Theorem). If [I' is a class of sets closed under finite intersections, and !?) is a Dynkin system containing Y, then !?) contains a(Y). A1.2. Topologies A topology 1111 on a space !!l is a class of subsets of !!l that is closed under arbitrary unions and finite intersections and that includes the empty set 0 and the whole space !!l; the members of 1111 are open sets, while their comple- 594 APPENDIX 1. A Review of Some Basic Concepts ments are closed sets. The pair (&r, Cl//) is a topological space. The closure of an arbitrary set A from &r, written A, is the smallest closed set (equivalently, the intersection of all closed sets) containing A. The interior of A, written A 0 , is the largest open set (equivalently, the union of all open sets) contained within A. The boundary of A, written oA, is the difference A\A0 • The following elementary properties of boundaries are needed in the discussion of weak convergence of measures. Proposition A1.2.1. (a) o(A u B) ~ oA u oB; (b) o(A n B) ~ oA u oB; (c) oAC = oA. A neighbourhood of the point x E &r with respect to the topology Cllt (or, more briefly, a 1111-neighbourhood of x) is an open set from Cl/t containing x. Cl/t is a Hausdorff or T2 topology if the open sets separate points, that is, if for x =F y, x and y possess disjoint neighbourhoods. A family of sets ff forms a base for the topology 1111 if every U E 1111 can be represented as a union of sets in ff, and ff ~ 1111. 1111 is second countable if it has a countable base, that is, if there exists a countable family ff with these properties. The topology Cl//1 generated by the base !F;_ has the same open sets as, and hence coincides with, the topology 11112 generated by the base ff,_ if every set in /#'1 can be represented as a union of sets in :#'2 , and vice versa. In this case the topological spaces (&r, Cl/t 1 ) and (&r, Cllt 2 ) are said to be equivalent. Given a topology Cl/t on &r, 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 1111 if, given any Cl//-neighbourhood of x, 1111x, there exists an integer N (depending on the neighbourhood in general) such that Xn E 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 over­ whelming emphasis is on metric topologies, where the open sets are defined in terms of a metric or distance function p( ·) satisfying for arbitrary x, y, z E &r the conditions (i) p(x, y) = p(y, x); (ii) p(x, y) ~ 0 and p(x, y) = 0 if and only if x = y; (iii) (triangle inequality) p(x, y) + p(y, z) ~ p(x, z). With respect to a given distance function p, the open sphere S(x, e) is the set {y: p(x, y) < e}, being defined for any e > 0. The metric topology generated by p 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 p(xn, x) --+ 0. A limit point y of a set A is a limit of a sequence of points Xn E A, with Xn =I= 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 &r with topology 1111 is metrizable if a distance A1.2. Topologies 595 function p can be found such that if/1 is equivalent to the metric topology generated by p. Two metrics on the same space X are equivalent if they each generate the same topology on X. A sequence of points {x": n ;;:,: 1} in a metric space is a Cauchy sequence if p(x", xm)--+ 0 as n, m--+ oo. The space is complete if every Cauchy sequence has a limit, that is, if for every Cauchy sequence {x"} there exists a point x EX such that p(x", x)--+ 0. A set ::0 is dense in X if, for every e > 0, every point in X can be approximated by points in ::0, that is, given x E X, there exists d E ::0 such that p(x, d) < e. The space X is separable if there exists a countable dense set, also called a separability set. If X is a separable metric space, the spheres with rational radii and centres on a countable dense set form a countable basis for the topology. Given two topological spaces (X1 , if/11 ) and (X2 , if/12 ), a mapping f( ·)from (X1 , if/1 1 ) into (X2 , if/12 ) is continuous if the inverse imagef-1 (U) of every open set U in if/12 is an open set in if/1 1 • If both spaces are metric spaces, the mapping is continuous if and only if for every x E X1 and every e > 0, there exists c5 > 0 such that p2 (f(x'),f(x)) < e whenever p 1 (x', x) < c5, where Pu p2 are the metrics in Xu X2 , respectively, a statement we can more loosely express by saying f(x')--+ f(x) whenever x' --+ x.
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