Measures and probabilities

Integration theory holds a prime position, whether in pure mathematics or in various fields of applied mathematics. It plays a central role in analysis; it is the basis of probability theory and provides an indispensable tool in mathe­ matical physics, in particular in quantum mechanics and statistical mechanics. Therefore, many textbooks devoted to integration theory are already avail­ able. The present book by Michel Simonnet differs from the previous texts in many respects, and, for that reason, it is to be particularly recommended. When dealing with integration theory, some authors choose, as a starting point, the notion of a measure on a family of subsets of a set; this approach is especially well suited to applications in probability theory. Other authors prefer to start with the notion of Radon measure (a continuous linear func­ tional on the space of continuous functions with compact support on a locally compact space) because it plays an important role in analysis and prepares for the study of distribution theory. Starting off with the notion of Daniell measure, Mr. Simonnet provides a unified treatment of these two approaches. This Book Is Intended To Be An Introductory, Yet Sophisticated, Treatment Of Measure Theory. It Should Provide An In-depth Reference For The Practicing Mathematician. It Is Hoped That Advanced Students As Well As Instructors Will Find It Useful. The First Part Of The Book Should Prove Useful To Both Analysts And Probabilists. One May Treat The Second And Third Parts As An Introduction To The Theory Of Probability, Or Use The Fourth Part As An Introduction To Analysis. The Treatment Is For The Most Part Self-contained. Other Than Familiarity With General Topology, Some Functional Analysis And A Certain Degree Of Mathematical Sophistication, Little Is Required For Profitable Reading Of This Text. At The End Of Each Chapter, Exercises Are Provided Which Are Designed To Present Some Additional Material And Examples. Michel Simonnet ; Foreword By Charles-michel Marle. Includes Indexes. Content: 1. Riesz Spaces -- 2. Measures on Semirings -- 3. Integrable and Measurable Functions -- 4. Lebesgue Measure on R -- 5. L[superscript p] Spaces -- 6. Integrable Functions for Measures on Semirings -- 7. Radon Measures -- 8. Regularity -- 9. Induced Measures and Product Measures -- 10. Radon-Nikodym Derivatives -- 11. Images of Measures -- 12. Change of Variables -- 13. Stieltjes Integral -- 14. The Fourier Transform in R[superscript k] -- 15. The Strong Law of Large Numbers -- 16. The Central Limit Theorem -- 17. Order Statistics -- 18. Conditional Probability -- 19. [mu]-Adequate Family of Measures -- 20. Radon Measures Defined by Densities -- 21. Images of Radon Measures and Product Measures -- 22. Operations on Regular Measures -- 23. Haar Measures -- 24. Convolution of Measures.