Download Integration on Infinite-Dimensional Surfaces and Its by A. Uglanov PDF

By A. Uglanov

It turns out tough to think, yet mathematicians weren't drawn to integration difficulties on infinite-dimensional nonlinear constructions as much as 70s of our century. a minimum of the writer isn't conscious of any e-book pertaining to this topic, even supposing as early as 1967 L. Gross pointed out that the research on endless­ 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 afterward. In these days the combination idea in endless­ dimensional linear areas was once basically built within the heuristic works of RP. Feynman [1], I. M. Gelfand, A. M. Yaglom [1]). The articles of J. Eells [1], J. Eells and ok. D. Elworthy [1], H. -H. Kuo [1], V. Goodman [1], the place the contraction of a Gaussian degree on a hypersurface, specifically, used to be equipped and the divergence theorem (the Gauss-Ostrogradskii formulation) was once proved, seemed basically at the start of the 70s. therefore 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 development challenge isn't really uncomplicated and has now not but been solved. A. V. Skorokhod [1] and the writer [6,10] provided assorted methods to one of these development. 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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Wnen, Y = YI el + ... + Ynen, Wi, Yi E R 1). /,F = >'Fzn, where >'F : Y -t Rl : >'F(Y) = Del .. /,(F X Wy). From the expression for AF it follows that the function >. (y, F) = >'F(Y) is a transitional measure from Y to X. L I(Ax x Y). /, ~ lTzn. Since Del ... Llx. 16) for all f3 ~ (}: . 2. /,Ixzn. 18) 51 SURFACE INTEGRALS where the functions rp, 'ljJ : Z -+ Rl are integrable with respect to the measure Il1lxzn. We put Ei = Ex X Eyi; l1i = Il1lx I1lj. 17) and j#i the Fubini theorem for any Ai E E i , t E 1"; we have ff t p(Ai X (-00, t)) = -00 rp(Xi' s) dl1i(Xi) dli(s).

3 be luifilled, the mea- sure vp, be a-bounded, and the lunction I : Z -+ L(E, F) be vp,-integrable. Then lor p,-almost all x the lunction y M I(x, y) is vex, ·)-integrable, the lunction x M f I(x, y) vex, dy) is equivalent (mod p,) to a p,-integrable one, y and, moreover, f = I dvp, z Proof. 3) I(x, y)v(x, dy) df-L(x). y I is vp,-integrable, then f 11/11 dlvp,1 < z 00. 4) 00. Firstly we prove the theorem for the case when Ivp,I(Z) < We set {x EX: Ivl(x, Y) = oo}, Xl X2 00. = {x EX: f III(x,y)lIlvl(x,dy) = oo}.

LIln)) and the Lebesgue Y theorem about a passage to the limit under the integral sign we obtain lim t-tO ~[f-L(A + ten) t lim! t-tO {L(A)] ften (Xn, Yn) dln(Yn) dJin(xn) ! Xn SXn (A) ! 23) SURFACE INTEGRALS 53 The induction application gives now the inclusion IL E MO:. 23). 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).

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