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The product-decomposability of probability measures on Abelian metrizable groups

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Krakowiak W.

Instytut Matematyczny Polskiej Akademii Nauk

Introduction.............................................................5 I. Preliminaries.........................................................6 1.1. Semigroups........................................7 1.2. Algebraic groups..................................7 1.3. Additive operators in Abelian groups and linear operators in linear spaces................................8 1.4. Abelian metrizable groups........................10 1.5. Locally compact Abelian groups...................13 1.6. Transformation groups............................15 1.7. Locally convex spaces............................16 1.8. The space $ℝ^∞$..................................18 II. Basic properties of probability measures............................19 2.1. Probability measures on metrizable spaces................................19 2.2. Probability measures on transformation groups............................20 2.3. Probability measures on Abelian metrizable groups........................22 2.4. Invariant subgroups of probability measures..............................26 III. Borel decomposability semigroups of probability measures...................................29 3.1. Additive measurable operators in Abelian metrizable groups...............29 3.2. Borel decomposability semigroups of probability measures.................33 3.3. Additive projections in Borel decomposability semigroups of probability measures.........................35 3.4. Additive projections in Borel decomposability semigroups of probability measures without idempotent factors....................................41 IV. Product-decomposability of probability measures.....................46 4.1. Basic definitions and results....................46 4.2. Gaussian measures in the sense of Gnedenko...............................47 4.3. Gaussian measures in the sense of Gnedenko without idempotent factors....49 4.4. Product-atoms in Borel decomposability semigroups of probability measures without idempotent factors.....55 4.5. Product-atomless probability measures without idempotent factors.........57 4.6. Canonical product-decomposition of probability measures..................60 V. Product-decomposability of probability measures on locally convex metrizable spaces..........61 5.1. Strong product-decomposability of probability measures on metrizable linear spaces.......................61 5.2. Infinitely divisible probability measures on locally convex metrizable spaces............................65 5.3. Gaussian measures on locally convex metrizable spaces....................67 5.4. Product-atomless probability measures on locally convex metrizable spaces................................71 5.5. Canonical product-decomposition and canonical strong product-decomposition of probability measures on locally convex metrizable spaces.........72 VI. Product decomposability of probability measures on LCA metrizable groups....................73 6.1. Initial results on probability measures..........73 6.2. Gaussian measures................................76 6.3. Product-atomless probability measures............78 6.4. Canonical product-decomposition of probability measures..................80 References..............................................................81 Index of symbols........................................................83 Subject index...........................................................85

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1 Krakowiak W.

1 1996

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The product-decomposability of probability measures on Abelian metrizable groups