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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Extra info for Integration on Infinite-Dimensional Surfaces and Its Applications
Example text
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).