Random and Vector Measures
The book is devoted to the structural analysis of vector and random (or both) valued countably additive measures, and used for integral representations of random fields. The spaces can be Banach or Frechet types. Special attention is given to Bochner's boundedness principle and Grothendieck's representation unifying and simplyfying stochastic integrations. Several stationary aspects, extensions and random currents as well as related multilinear forms are analyzed, whilst numerous new procedures and results are included, and many research areas are opened up which also display the geometric aspects in multi dimensions.
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1 Introduction and Motivation
2 Second Order Random Measures and Representations
3 Random Measures Admitting Controls
5 More on Random Measures and Integrals
6 Martingale Type Measures and Their Integrals
7 Multiple Random Measures and Integrals
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adjoint algebra analysis applications Banach space bilinear form bimeasure Bo(R Bochner Borel sets bounded linear Chapter classical condition consider continuous controlling measure covariance decomposition defined denoted dimensional disjoint Dunford-Schwartz exists extended filtration finite Vitali Fréchet Fréchet space functions f Gaussian given harmonizable Hence Hilbert space implies inequality integral representation integrands Karhunen Lá(P LCA group Lebesgue Lebesgue measure locally compact Lp(P mapping martingale Math measurable space measure µ metric MT-integration nontrivial norm o-additive obtain orthogonally valued p-stable positive definite probability space Proof Proposition random field random measure random variables relative result satisfies scalar Section semi-martingale sequence signed measure simple functions ſş stationary stochastic integral subspace Theorem theory tion topology unique vector measure verified weakly well-defined