The series "Grundlehren der mathematischen Wissenschaften" (subtitled Comprehensive Studies in Mathematics), Springer's first series in higher mathematics, was founded by Richard Courant in 1920. The Grundlehren were conceived as a series of modern textbooks. This character is obvious in the first 60-70 volumes, but the series underwent a number of significant changes after World War II. Outwardly, the change was in language. Whereas the overwhelming majority of the first 100 volumes were published in German, the following volumes are nearly all in English. A more important change concerns the contents of the books. The original objective of the Grundlehren had been to lead students to the principal results and to recent research questions in a single relatively elementary and easily readable book.

Multidimensional Diffusion Processes (Graduate Texts in Mathematics)

Science & Math

Daniel W. Stroock

Mathematics

Springer-Verlag Telos

The choice of topics included in this book, as well as the presentation of those topics, has been guided by the author's experience in teaching this material to classes consisting of advanced graduate students who are not concentrating in mathematics. This book contains an introduction to the modern theory of integration with a strong emphasis on the case of Lebesgue's measure for (Rn and eye toward applications to real analysis and probability theory. Following a brief review of the classical Riemann theory in Chapter I, the details of Lebesgue's construction are given in Chapter Ii, which also contains a derivation of the transformation properties of Lebesgue's measure under linear maps. Chapter Iii is devoted to Lebesgue's theory of integration of real-valued functions on a general measure space. Besides the basic convergence theorems, this chapter introduces product measures and Fubini's Theorem. In Chapter Iv, various topics having to do with the transformation properties of measures are derived. These include: the representation of general integrals in terms of Riemann integrals with respect to the distribution function, polar coordinates, Jacobi's transformation formula and finally the introduction of surface measure followed by a proof of the Divergence Theorem. A few of the basic inequalitites of measure theory are derived in Chapter V. In particular, the inequalities of Jensen, Minkowski and HÖLder are presented. Finally, Chapter Vi starts with the Daniell integral and its applications to the CarathÉOdory Extension and Riesz Representation Theorems. It closes with Von Neumann's derivation of the Radon-Nikodym Theorem.

A Concise Introduction to the Theory of Integration

Birkhauser

Essentials of Integration Theory for Analysis (Graduate Texts in Mathematics, 262)

<p>‘A Concise Introduction to the Theory of Integration’ was once a best-selling Birkhäuser title which published 3 editions. This manuscript is a substantial revision of the material. Chapter one now includes a section about the rate of convergence of Riemann sums. The second chapter now covers both Lebesgue and Bernoulli measures, whose relation to one another is discussed. The third chapter now includes a proof of Lebesgue's differential theorem for all monotone functions. This is a beautiful topic which is not often covered. The treatment of surface measure and the divergence theorem in the fifth chapter has been improved. Loose ends from the discussion of the Euler-MacLauren in Chapter I are tied together in Chapter seven. Chapter eight has been expanded to include a proof of Carathéory's method for constructing measures; his result is applied to the construction of Hausdorff measures. The new material is complemented by the addition of several new problems based on that material.</p>

Springer

<p>This book covers the basics of modern probability theory. It begins with probability theory on finite and countable sample spaces and then passes from there to a concise course on measure theory, which is followed by some initial applications to probability theory, including independence and conditional expectations. The second half of the book deals with Gaussian random variables, with Markov chains, with a few continuous parameter processes, including Brownian motion, and, finally, with martingales, both discrete and continuous parameter ones. The book is a self-contained introduction to probability theory and the measure theory required to study it.</p>

Mathematics of Probability (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 149)

American Mathematical Society

MATHEMATICS OF PROBABILITY

Orient Blackswan

Lectures on Topics in Stochastic Differential Equations: Lectures delivered at the Indian Institute of Science, Bangalore under the T.I.F.R.-I.I.SC. ... Lectures on Mathematics and Physics)

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Books by Daniel W. Stroock