Measures and Probabilities

ISBN-13: 9780387946443 / Angielski / Miękka / 1996 / 510 str.

Michel Simmonet; M. Simonnet; Michel Simonnet

Opis

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.

Spis treści

I Integration Relative to Daniell Measures.- 1 Riesz Spaces.- 1.1 Ordered Groups.- 1.2 Riesz Spaces.- 1.3 Order Dual of a Riesz Space.- 1.4 Daniell Measures.- 2 Measures on Semirings.- 2.1 Semirings, Rings, and ?-Rings.- 2.2 Measures on Semirings.- 2.3 Lebesgue Measure on an Interval.- 3 Integrable and Measurable Functions.- 3.1 Upper Integral of a Positive Function.- 3.2 Convergence Theorems.- 3.3 Integrable Sets.- 3.4 ?-Measurable Spaces.- 3.5 Measurable Mappings.- 3.6 Essentially Integrable Mappings.- 3.7 Upper and Lower Integrals.- 3.8 Atoms.- 3.9 Prolongations of ?.- 4 Lebesgue Measure on R.- 4.1 Base-b Expansions of a Real Number.- 4.2 The Cantor Singular Function.- 4.3 Example of a Nonmeasurable Set.- 5 Lp Spaces.- 5.1 Definition of Lp Spaces.- 5.2 Convergence Theorems.- 5.3 Convergence in Measure.- 5.4 Uniformly Integrable Sets.- 6 Integrable Functions for Measures on Semirings.- 6.1 Measurability.- 6.2 Complements on the Lp Spaces.- 6.3 Measures Defined by Masses.- 6.4 Prolongations of a Measure.- 7 Radon Measures.- 7.1 Locally Compact Spaces.- 7.2 Radon Measures.- 7.3 Regularity of Radon Measures.- 7.4 Lusin Measurable Mappings.- 7.5 Atomic Radon Measures.- 7.6 The Riemann Integral.- 7.7 Weak Convergence.- 7.8 Tight Sequences.- 8 Regularity.- 8.1 Regular Measures.- II Operations on Measures Defined on Semirings.- 9 Induced Measures and Product Measures.- 9.1 Measure Induced on a Measurable Set.- 9.2 Fubini’s Theorem.- 9.3 Lebesgue Measure on Rk.- 10 Radon-Nikodym Derivatives.- 10.1 Sums of Measures.- 10.2 Locally Integrable Functions.- 10.3 The Radon-Nikodym Theorem.- 10.4 Combination of Operations on Measures.- 10.5 Duality of Lp Spaces.- 10.6 The Yosida-Hewitt Decomposition Theorem.- 11 Images of Measures.- 11.1 ?-Suited Pairs.- 11.2 Infinite Product of Measures.- 11.3 Change of Variable.- 11.4 Elements of Ergodic Theory.- 12 Change of Variables.- 12.1 Differentiation in Rk.- 12.2 The Modulus of an Automorphism.- 12.3 Change of Variables.- 12.4 Polar Coordinates.- 13 Stieltjes Integral.- 13.1 Functions of Bounded Variation.- 13.2 Stieltjes Measures.- 13.3 Line Integrals.- 13.4 The Lebesgue Decomposition of a Function.- 13.5 Upper and Lower Derivatives.- 14 The Fourier Transform in Rk.- 14.1 Measures in Rk.- 14.2 Distribution Functions.- 14.3 Covariance Matrix.- 14.4 The Fourier Transform.- 14.5 Normal Laws in Rn.- III Convergence of Random Variables; Conditional Expectation.- 15 The Strong Law of Large Numbers.- 15.1 Convergence in Probability.- 15.2 Independence of Random Variables.- 15.3 An Example of Independent Random Variables.- 15.4 The One-Sided Shift Transformation.- 15.5 Borel’s Normal Number Theorem.- 16 The Central Limit Theorem.- 16.1 Convergence in Law.- 16.2 The Lindeberg Theorem.- 16.3 The Central Limit Theorem.- 17 Order Statistics.- 17.1 Definition of the Order Statistics.- 17.2 Convergence of the Empirical Median.- 18 Conditional Probability.- 18.1 Conditional Expectation.- 18.2 The Converse of the Mean-Value Theorem.- 18.3 Jensen’s Inequality.- 18.4 Conditional Expected Value Given a Random Variable.- 18.5 Conditional Law of Y Given X.- 18.6 Computation of Conditional Laws.- 18.7 Existence of Conditional Laws when G = Rk.- IV Operations on Radon Measures.- 19 ?-Adequate Family of Measures.- 19.1 Induced Radon Measure.- 19.2 ?-Dense Families of Compact Sets.- 19.3 Sums of Radon Measures.- 19.4 ?-Adequate Families.- 19.5 ?-Adapted Pairs.- 20 Radon Measures Defined by Densities.- 20.1 Integration with Respect to Induced Measures.- 20.2 Radon Measures with Base ?.- 20.3 The Radon-Nikodym Theorem.- 20.4 Duality of Lp Spaces.- 21 Images of Radon Measures and Product Measures.- 21.1 Images of Radon Measures.- 21.2 Decomposition of a Measure in Slices.- 21.3 Product of Radon Measures.- 22 Operations on Regular Measures.- 22.1 Some Operations on Regular Measures.- 22.2 Baire Sets.- 22.3 Product of Regular Measures.- 22.4 Change of Variable Formula.- 23 Haar Measures.- 23.1 Invariant Measures.- 23.2 Existence and Uniqueness of Left Haar Measure.- 23.3 Modular Function on G.- 23.4 Relatively Invariant Measures on a Group.- 23.5 Homogeneous Spaces.- 23.6 Integration with Respect to ?#.- 23.7 Reconstitution of ?#/?.- 23.8 Quasi-Invariant Measures on Homogeneous Spaces.- 23.9 Relatively Invariant Measures on G/H.- 23.10 Haar Measure on SO(n + 1,R).- 23.11 Haar Measure on SH(n,R).- 24 Convolution of Measures.- 24.1 Convolvable Measures.- 24.2 Convolution of a Measure and a Function.- 24.3 Convolution of a Measure and a Continuous Function.- 24.4 Convolution of ? ? M(G, C) and f ? $$\overline {{\mathcal^{\text}}}$$(?).- 24.5 Convolution and Transposition.- 24.6 Convolution of Functions on a Group.- 24.7 Regularization.- 24.8 Definition of Gelfand Pair.- Symbol Index.