Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Theory of fuzzy integrals and its applications

Real and Complex Analysis

The generic term fuzzy quantifier is employed in this paper to denote the collection of quantifiers in natural languages whose representative elements are: several, most, much, not many, very many, not very many, few, quite a few, large number, small number, close to five, approximately ten, frequently, etc. In our approach, such quantifiers are treated as fuzzy numbers which may be manipulated through the use of fuzzy arithmetic and, more generally, fuzzy logic.

A concept which plays an essential role in the treatment of fuzzy quantifiers is that of the cardinality of a fuzzy set. Through the use of this concept, the meaning of a proposition containing one or more fuzzy quantifiers may be represented as a system of elastic constraints whose domain is a collection of fuzzy relations in a relational database. This representation, then, provides a basis for inference from premises which contain fuzzy quantifiers. For example, from the propositions “Most U's are A's” and “Most A's are B's,” it follows that “Most2 U's are B's,” where most2 is the fuzzy product of the fuzzy proportion most with itself.

The computational approach to fuzzy quantifiers which is described in this paper may be viewed as a derivative of fuzzy logic and test-score semantics. In this semantics, the meaning of a semantic entity is represented as a procedure which tests, scores and aggregates the elastic constraints which are induced by the entity in question.

A computational approach to fuzzy quantifiers in natural languages

Fuzzy topological spaces and fuzzy compactness

A Comparison of Different Compactness Notions in Fuzzy Topological Spaces

In the first paragraph we study filters in the lattice IX, where I is the unitinterval and X an arbitrary set. The main result of this section is a characterization of minimal prime filters in IX containing a given filter in IX by means of ultrafilters on X. In the second paragraph we apply the results of the previous section to define convergence in a fuzzy topological space which enables us to characterize fuzzy compactness and fuzzy continuity.

Convergence in fuzzy topological spaces

Symbols Preface 1. Approach Spaces 2. Topological Approach Spaces 3. Metric Approach Spaces 4. Uniform Approach Spaces 5. Canonical Examples 6. Approach Properties 7. Completion 8. Compactification Appendix A Appendix B Index Bibliography

Robert Lowen

Papers

Fuzzy topological spaces and fuzzy compactness

A Comparison of Different Compactness Notions in Fuzzy Topological Spaces

Convergence in fuzzy topological spaces

Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad

Fuzzy Uniform Spaces