Probability Distributions on Banach Spaces

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Klappentext
Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non­ trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Inhalt
I. Measurability and Measures.- 1. ?-algebras and measurable mappings in metric space.- 2. ?-algebras in Banach spaces.- 3. Probability measures on topological spaces.- 4. The semigroup of probability measures.- 5. Invariant and quasi-invariant measures. Scalarly non-degenerate measures.- Supplementary comments.- II. Measures and Random Elements. Weak and Strong Orders.- 1. Random elements.- 2. Weak and strong orders of random elements.- 3. The expectation.- 4. Conditional expectations, martingales and connections with the Radon-Nikodym Property.- Supplementary comments.- III. Covariance Operators.- 1. Operators mapping spaces into their duals.- 2. Covariance operators.- Supplementary comments.- IV. Characteristic Functionals.- 1. Positive-definite functions.- 2. Definition and general properties of characteristic functionals.- 3. Characteristic functionals and weak convergence.- 4. Bochner's theorem.- Supplementary comments.- V. Sums of Independent Random Elements.- 1. Independent random elements.- 2. Series of independent random elements.- 3. Integrability of sums and the mean convergence of random series.- 4. Comparison of random series.- 5. Some special series.- 6. Random series in spaces which do not contain c0.- Supplementary comments.- VI. Topological Description of Characteristic Functionals and Cylindrical Measures.- 1. Sazonov's theorem and related topics.- 2. Necessary and sufficient topologies. Spaces with the Sazonov property.- 3. Cylindrical measures.- 4. The case of locally convex spaces. Minlos' theorem.- 5. Radonifying operators.- Supplementary comments.

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