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

What is modal logic

Merging fundamental concepts of analysis and recursion theory to a new exciting theory, this book provides a solid fundament for studying various aspects of computability and complexity in analysis. It is the result of an introductory course given for several years and is written in a style suitable for graduate-level and senior students in computer science and mathematics. Many examples illustrate the new concepts while numerous exercises of varying difficulty extend the material and stimulate readers to work actively on the text.

Computable Analysis : An Introduction

Computability on subsets of metric spaces

A Tutorial on Computable Analysis

Effective properties of sets and functions in metric spaces with computability structure

Computability Structures on Metric Spaces.

We speculate on computational aspects of certain discontinuous functions by taking the Gau ian function [x] as a typical example. An algorithm how to compute [x] for a single computable real number is rst described, followed by a remark that [x] does not necessarily preserve sequential computability. Second, [x] is studied in the light of the notions of upper semi-computability and of limiting computation. Then two Fr echet spaces, Rand L1loc(R) , in which some discontinuous functions will become computable, will be taken up.

Computability aspects of some discontinuous functions

We develop the theory of the computability structure and some notions of computable functions on a uniform topological space, and will apply the results to some real functions which are discontinuous in the Euclidean space.

Some Properties of the Effective Uniform Topological Space

In this article, we discuss the Fine computability and the effective Fine convergence for functions on [0, 1) with respect to the Fine metric as the beginning of the effective Walsh-Fourier analysis. First we treat classically the Fine continuity and the Fine convergence. Next, we prove that Fine computability does not depend on the choice of an effective separating set. Subsequently, we propose a notion of effective Fine convergence for a sequence of functions. We prove that the limit of an effectively Fine continuous sequence of functions and the limit of a Fine computable sequence of functions under this effective Fine convergence is effectively Fine continuous and Fine computable respectively. We also investigate some properties of Fine computable functions through examples. Especially, we extend the result of Brattka, which asserts the existence of a Fine computable but not locally uniformly Fine continuous function. Finally, we treat other examples of Fine computable functions.

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Mariko Yasugi

Papers

Effective properties of sets and functions in metric spaces with computability structure

Computability Structures on Metric Spaces.

Computability aspects of some discontinuous functions

Some Properties of the Effective Uniform Topological Space

Fine Computable Functions and Effective Fine Convergence.