lEt moi, .... si j'avait Sll comment en revenir, One service mathematics has rendered the human race. It has put common sense back je n'y serais point aile:' where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded 0- sense'. The series is divergent; therefore we may be Eric T. Bell able to do something with it. o. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non­ linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com­ puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d 'e1re of this series.

I. Measures and quasimeasures. Integration.- 1. Realvalued measures on algebras of sets.- 2. Cylinder sets and cylindrical functions.- 3. Quasimeasures. Integration.- 4. Supplement: Some notions related to the topology of linear spaces.- 5. Chapter I: Supplementary remarks and historical comments.- II. Gaussian measures in Hilbert space.- 1. Gaussian measures in finite-dimensional spaces.- 2. Gaussian measures in Hilbert space.- 3. Measurable linear functionals and operators.- 4. Absolute continuity of Gaussian measures.- 5. Fourier-Wiener transformation.- 6. Complexvalued Gaussian quasimeasures.- 7. Chapter II: Supplementary re marks and historical comments.- III. Measures in linear topological spaces.- 1. ?-additivity conditions for nonnegative cylindrical measures in the space X' dual to a locally convex space X.- 2. Sequences of Radon measures.- 3. Chapter III: Supplementary remarks and historical comments.- IV. Differentiable measures and distributions.- 1. Differentiable functions, differentiable expressions.- 2. Differentiable measures.- 3. Distributions and generalized functions.- 4. Positive definiteness. Quasi-invariant distributions and bidistributions.- 5. Chapter IV: Supplementary remarks and historical comments.- V. Evolution differential equations.- 1. Weak solutions of evolution equations.- 2. Second order equations with variable coefficient.- 3. Chapter V: Supplementary remarks and historical comments.- VI. Integration in path space.- 1. Markov quasimeasures.- 2. Evolution families of operators.- 3. Linear evolution families and functional integrals.- 4. Nonlinear evolution families, and integrals in branching path space.- 5. Chapter VI: Supplementary remarks and historical comments.- VII. Probabilistic representations of solutions of parabolicequations and systems.- 1. Wiener process. Stochastic integrals.- 2. Stochastic differential equations.- 3. Operator multiplicative functionals and the evolution families determined by them.- 4. The Cauchy problem for second order parabolic systems.- 5. Chapter VII: Supplementary remarks and historical comments.- VIII. Smooth measures.- 1. Admissible operators.- 2. Admissibility of differential operators.- 3. Absolute continuity of smooth measures.- 4. Maps of spaces and admissible operators.- 5. Biorthogonal systems in L2(X, ?).- 6. Chapter VIII: Supplementary remarks and historical comments.- Supplement to chapters IV-V.- 1. Essentially infinite-dimensional elliptic operators.- 2. Properties of essentially infinite-dimensional elliptic operators, and solutions of the corresponding Cauchy problem.- 3. Existence of solutions of the Cauchy problem.- 4. Supplementary remarks and historical comments.- Supplement to chapter VII.- 1. Supplementary remarks and historical comments.