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Functions --- Topology --- Set functions

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Stochastic processes --- Functional analysis --- Martingales (Mathematics) --- Set functions. --- Martingales (Mathematics).

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Functional analysis --- Differentiable functions --- Interval functions --- Functions, Interval --- Set functions --- Functions, Differentiable --- Functions of real variables --- Fonctions différentiables --- Fonctions différentiables.

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Operator theory and functional analysis have a long tradition, initially being guided by problems from mathematical physics and applied mathematics. Much of the work in Banach spaces from the 1930's onwards resulted from investigating how much real (and complex) variable function theory might be extended to futions taking values in (function) spaces or operators acting in them. Many of the first ideas in geometry, basis theory and the isomorphic theory of Banach spaces have vector measure-theoretic origins and can be credited (amongst others) to N. Dunford, I.M. Gelfand, B.J. Pettis and R.S. Phillips. Somewhat later came the penetrating contributions of A.Grothendieck, which have pervaded and influenced the shape of functional analysis and the theory of vector measures/integration ever since. Today, each of the areas of functional analysis/operator theory, Banach spaces, and vector measures/integration is a strong discipline in its own right. However, it is not always made clear that these areas grew up together as cousins and that they had, and still have, enormous influences on one another. One of the aims of this monograph is to reinforce and make transparent precisely this important point.

Set functions. --- Linear operators. --- Function spaces. --- Functional analysis. --- Integral operators. --- Spaces, Function --- Functional analysis --- Linear maps --- Maps, Linear --- Operators, Linear --- Operator theory --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Functions, Set --- Functions of real variables --- Set theory --- Operators, Integral --- Integrals --- Global analysis (Mathematics). --- Operator theory. --- Analysis. --- Operator Theory. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical analysis. --- Analysis (Mathematics). --- 517.1 Mathematical analysis --- Mathematical analysis

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Symmetries and invariance principles play an important role in various branches of mathematics. This book deals with measures having weak symmetry properties. Even mild conditions ensure that all invariant Borel measures on a second countable locally compact space can be expressed as images of specific product measures under a fixed mapping. The results derived in this book are interesting for their own and, moreover, a number of carefully investigated examples underline and illustrate their usefulness and applicability for integration problems, stochastic simulations and statistical applications.

Measure theory. --- Probability measures. --- Measure theory --- Probability measures --- Set functions --- Mathematical Theory --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Topological groups. --- Lie groups. --- Numerical analysis. --- Statistics . --- Measure and Integration. --- Topological Groups, Lie Groups. --- Numerical Analysis. --- Statistical Theory and Methods. --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Econometrics --- Mathematical analysis --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Groups, Topological --- Continuous groups --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra)