By A. Uglanov

It turns out demanding to think, yet mathematicians weren't drawn to integration difficulties on infinite-dimensional nonlinear buildings as much as 70s of our century. at the least the writer isn't really conscious of any e-book bearing on this subject, even supposing as early as 1967 L. Gross pointed out that the research on countless dimensional manifolds is a box of study with relatively wealthy possibilities in his classical paintings [2. This prediction was once brilliantly proven afterwards, yet we will go back to this in a while. In these days the mixing thought in countless dimensional linear areas was once primarily constructed within the heuristic works of RP. Feynman [1], I. M. Gelfand, A. M. Yaglom [1]). The articles of J. Eells [1], J. Eells and okay. D. Elworthy [1], H. -H. Kuo [1], V. Goodman [1], the place the contraction of a Gaussian degree on a hypersurface, specifically, was once outfitted and the divergence theorem (the Gauss-Ostrogradskii formulation) was once proved, seemed in simple terms before everything of the 70s. for this reason a Gaussian specificity was once crucial and it was once even mentioned in a later monograph of H. -H. Kuo [3] that the skin degree for the non-Gaussian case building challenge isn't uncomplicated and has no longer but been solved. A. V. Skorokhod [1] and the writer [6,10] provided varied techniques to one of these building. another ways have been provided later through Yu. L. Daletskii and B. D. Maryanin [1], O. G. Smolyanov [6], N. V.

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3). 1: B = L{Rl, B}). We denote by M(X, Y, B) the collection of all transitional measures. The set M{X, Y, [0,00]) is defined analogously. For CHAPTER 1 22 f-l E M(X, B), v E M(X, Y, B) and bE B* we use the notation bf-l : ~ -+ Rl : A I-t (b, f-l(A)); bv : X X ~y -+ Rl : (x, A) I-t (b, v(x, A)); Ivl : X X ~y -+ [0,00] : (x, A) I-t Ivl(x, ·)(A). For A E ~z and x E X let Sx(A) = {y E Y : (x, y) E A}. , [1], Ch. 4, and Neveu [1], Ch. III, §2). 1. A CASE OF VECTOR TRANSITIONAL AND SCALAR INITIAL MEASURES In this subsection f-l E M(X, Rl), v E M(X, Y, B).

4. 2. For the measures p, a defined on some a-algebra, p ~ a we denote by dp/da the equivalence class of the RadonNikodym densities of the measure p with respect to the measure a (so the notations f E dp/da and p = fa are equivalent). Let f : Z ---+ Rl is a funcion, z = (Xi, Yi) Xi E Xi, Yi E Yi. We consider that the derivative DeJ(Xi, Yi) exists if the funcion f(Xi,') : Yi ---+ Rl is Ii-equivalent to some function g : Yi ---+ Rl differentiable at the point Yi; in this case we put DeJ(z) = Deig(Yi).

L E M(Z, B). Proof. Let b E B*. L). 2 (with v E M(X, Y, B) replaced by v E M(X, Y, Rl) and f. L is weakly countably additive, hence countably additive. L1 : ~z -+ Rl : A is countably additive. L1 f--i 30 CHAPTER 1 Proof. If IIlI (X) < Il replaced by IIll) . :x and 11l1(Xn ) let Zn = z;t < 00. UZ;;:- Then villi 00 = U X n , Xl n=l C X 2 C ... :z X Y; 00 U Z~. n=l for 11l1-almost all x EX, and the countable additivity of villi follows from the Lebesgue theorem. , not a-bounded) measure Il. :z, Al C A2 C ...