1. General properties of sets. 2. Measure of sets. 3. Concept of Lebesgue integral. 4. Integrals before Lebesgue. 5. Measurable functions. 6. Lebesgue integrals of bounded functions. 7. Passage to limit under integral sign. 8. Integrals of unbounded functions. 9. Sets and functions measurable (B). 10. Functions representable analytically. Classes of Baire. 11. Reduction of multiple integrals to iterated integrals. 12. Geometric interpretation of Lebesgue integral. 13 Integrals analogous to Lebesgue integrals. 14. Functions of bounded variation. 15. Indefinite L-integral. Derived numbers. Differentiation of functions of bounded variation. 16. Differentiation of indefinite integrals. 17. Characterization of indefinite integrals. 18. Integration of a derivative. Rule of Barlow. 19. Some properties of the L-integral. 20. General integral of Denjoy, or totalization. 21. Perron integral. 22. Stieltjes integral. 23. Linear functionals. 24. Stieltjes-Lebesgue integral, and integration in abstract spaces.
1.2. Review by: T H Hildebrandt.
Mathematical reviews MR0012331 (7,11e).
This monograph gives a clear and concise exposition of the theory of the Lebesgue integral, together with a brief introduction to the Denjoy, Perron and Stieltjes integrals. For difficult proofs, references are given to the original articles or other treatments of integration.
Mathematical reviews MR0012702 (7,61g).
This publication gives an account of some of the properties of Dirichlet series in the complex domain and their extension to Laplace-Stieltjes integrals. It antedates the recent publications of Mandelbrojt, and apparently Widder's book [The Laplace Transform, 1941] was not available to the author, so that some recent results are not included. Among the author's own contributions are studies on overconvergence, the determination of singularities, and, in the case of Dirichlet series, the question of analytic continuation by reordering.
Mathematical reviews MR0014504 (7,294e).
This appears to be a revision of the author's lectures [Lectures on the theory of the analytic continuation of Dirichlet series (Spanish)]. ... Some additional topics are presented. These include a discussion of the class of all Laplace-Stieltjes transforms as a complete metric space, the distance between two elements being eCif the abscissa of convergence of their difference is C, elements at zero distance being identified. The author also discusses the approximation of analytic functions by exponential polynomials.
Mathematical reviews MR0013788 (7,200b).
In these lectures (written in collaboration with L Vigil) the author covers a wide variety of topics in the representation of analytic functions of a complex variable: Runge's theorems; analytic continuation by overconvergence and by rearrangement (he constructs, among other examples, a "universal" series of polynomials which can be rearranged to converge uniformly to any prescribed analytic function in any desired region); Mittag-Leffler, Borel and Painlevé expansions; analytic continuation by summation of series; representation of functions by Laplace integrals and by Dirichlet, factorial, interpolation and Lambert series.
Mathematical reviews MR0029437 (10,603b).
Lectures given by the author in 1946-47, edited in collaboration with J Béjar, T Iglesias and M E Ríos.
Mathematical reviews MR0043410 (13,259b).
An expository survey of the concepts of measure and integral, the Kolmogorov identification of probability with measure theory, probability distributions, and the Fisher-Neyman-Pearson theories of estimation and testing hypotheses.
Introduction. Statistical tables and graphical representations. Frequency and probability. Sample and universe. Reduction of statistical data. Some distributions. Multi-dimensional statistical variables. Two-dimensional random variables. Introduction to sampling theory. Sampling distributions of some statistics. Intuitive introduction of tests and comparisons of statistical hypotheses.
8.2. Review by: Paul Halmos.
Journal of the American Statistical Association 48 (261) (1953), 154-155.
This is a charming and elementary book that fulfils, within the limits the author sets for it, the promise of the title. The mathematical level of the book is not advanced; nothing more high-powered than integral calculus is ever used. ... Although at times the definitions are a little vague, the large number of examples, complete with detailed tables and graphs, is likely to give the reader a sound intuitive grasp of the subject. A notable feature of the book (emphasised by Herman Wold in his preface) is the number, variety, and interest of the exercises.
8.3. Review by: H Chernoff.
Econometrica 26 (3) (1958), 479-480.
The author sets himself the task of writing a textbook on statistics for students with rather heterogeneous backgrounds. No prior knowledge of statistics is assumed and acquaintance with calculus is considered sufficient. In the first twelve chapters the material covered consists of the elements of a rather old-fashioned course on mathematical statistics. He discusses statistical tables and graphical representation, elementary probability theory, reduction of statistical data with mean, mode, median, and standard deviation, basic distributions, an introduction to sampling to estimate populations, and testing hypotheses from the significance level point of view. In Chapters 13 through 17, the level of sophistication of the statistical ideas is increased considerably. ... Chapters 18 and 19 are the most modern, introducing decision theory and sequential analysis. The remaining chapters cover non-parametric analysis, a "theory of errors," analysis of variance, design of experiments, regression and correlation, sampling from finite populations, time series, and stochastic processes. These are followed by an appendix on operations research with industrial and military applications. On the whole the author has shown excellent taste in the mathematical aspects of this book. The proofs he included are careful and rigorous and, what is equally important, neat and brief. Pedagogically, his explanations of statistical ideas are well motivated and well illustrated with a great deal of economy in the use of space and examples. References to more detailed and advanced results are given for the benefit of the more sophisticated students.
Estimation of parameters. Methods for the formation of estimators. Confidence intervals. Tests of statistical hypotheses. Decision functions. Sequential analysis. Estimation of distributions without parameters. Applications to the measurement of physical magnitudes; theory of errors; quality control. Analysis of variance. Design of experiments. Regression problems; multidimensional analysis. Finite populations. Time series. Stochastic processes. Appendix: Operations research.
9.2. Review by: S Vajda.
Journal of the Royal Statistical Society. Series A (General) 118 (1) (1955), 110-111.
The author of this book is the director of a school of statistics which has been organised in the University of Madrid since the last war. The word "school" should, in this context, be understood to mean a teaching establishment rather than a research group, and it is of interest to see how energetically Professor Rios assists his students in catching up with that statistical knowledge to which British and American scholars have, as yet, made significantly more contributions than his fellow countrymen. The volume now under review is the second part of a course in mathematical statistics. It deals with a number of subjects not often found in textbooks. The presentation of the theory of estimation and of testing hypotheses follows familiar lines, as do the chapters on quality control, design of experiments, multivariate analysis and on small samples. But there are also chapters on decision functions, sequential analysis, non-parametric distributions, time series and stochastic processes.
9.3. Review by: H P Mulholland.
Mathematical reviews MR0043410 (13,259b).
The author has produced an up-to-date introductory textbook (with exercises) that includes a wide survey of modern statistical methods. It is characteristic of his approach that Kolmogorov's axioms for probability make their first appearance in the last chapter. The book is extremely readable, partly owing to careful explanation (often with diagrams) of the ideas involved, and partly to omission of the harder mathematical proofs. Many results are merely stated as holding "under very general conditions".
(1) Types of decision problems; (2) Individual decisions in a stochastic environment; (3) Experiments and applications of the von Neumann utility theory; (4) Decisions under uncertainty; (5) Randomised decisions, admissibility; (6) Risk and uncertainty; (7) Models; (8) Utility for sets of goods and multicriteria; (9) Utility for investments; (10) Collective decisions.
10.2. Review by: Alan Harding.
Operational Research Quarterly (1970-1977) 28 (4.2) (1977), 1024-1025.
"The publication in 1943 of the treatise of Von Neumann, with its fundamental contributions to the theory of games and to the theory of utility, marks the point of departure for the application of scientific method to problems of decision ..." What a first sentence: back to fundamentals with a vengeance. Von Neumann's masterpiece is huge, thinks as it goes (and never mind the blind alleys) and is profoundly original. The present volume, whilst acknowledging its debt to Von Neumann, is comparatively short, beautifully clear and contains no new material. It reviews work on decision analysis from the game theoretic viewpoint, mostly up to the early 1970s, with some later additions. The author has clearly read the many works he cites, and this overview is one of the valuable aspects of the book. Ten chapters take you in logical steps from "individual decisions in a probabilistic environment" up to "utility in a multistage environment" and "collective decisions." It includes also a reasonably full treatment of Bayesian methods and is illustrated with simple examples.
10.3. Review by: Gerhard Tintner.
International Statistical Review/Revue Internationale de Statistique 46 (3) (1978), 322.
This is an excellent simple introduction to decision theory. ... . The theory is illustrated with the help of many well chosen examples. The notes and problems at the end of each chapter contribute to the value of this introduction.