Structural Aspects in the Theory of Probability: A Primer in Probabilities on Algebraic - Topological Structures -- Preface -- Contents -- 1 Probability Measures on Metric Spaces -- 1.1 Tight measures -- 1.2 The topology of weak convergence -- 1.3 The Prohorov theorem -- 1.4 Convolution of measures -- 2 The Fourier Transform in a Banach Space -- 2.1 Fourier transforms of probability measures -- 2.2 Shift compact sets of probability measures -- 2.3 Infinitely divisible and embeddable measures -- 2.4 Gauss and Poisson measures -- 3 The Structure of Infinitely Divisible Probability Measures -- 3.1 The Ito-Nishio theorem -- 3.2 Fourier expansion and construction of Brownian motion -- 3.3 Symmetric Levy measures and generalized Poisson measures -- 3.4 The Levy-Khintchine decomposition -- 4 Harmonic Analysis of Convolution Semigroups -- 4.1 Convolution of Radon measures -- 4.2 Duality of locally compact Abelian groups -- 4.3 Positive definite functions -- 4.4 Positive definite measures -- 5 Negative Definite Functions and Convolution Semigroups -- 5.1 Negative definite functions -- 5.2 Convolution semigroups and resolvents -- 5.3 Levy functions -- 5.4 The Levy-Khintchine representation -- 6 Probabilistic Properties of Convolution Semigroups -- 6.1 Transient convolution semigroups -- 6.2 The transience criterion -- 6.3 Recurrent random walks -- 6.4 Classification of transient random walks -- Appendices -- A Topological groups -- B Topological vector spaces -- C Commutative Banach algebras -- Selected References -- Symbols -- Index.