Finally, we are finding behind a first monography related to the integration problems on infinite-dimensional nonlinear structures.
After the heuristic works of R. P. Feynman (1948) and I. M. Gelfand, A. M. Yaglom (1956) where the integration theory in infinite-dimensional linear spaces was essentially developed, the works of J. Eells [“Integration on Banach manifolds”, Proc. 13th Biennial Semin. Can. Math. Congr. Differ. Topol. Differ. Geom. Appl. 1, 41-49 (1971/1972; Zbl 0268.58003)], J. Eells and K. D. Elworthy [“Wiener integration on certain manifolds”, Probl. nonlinear Anal. 67-94 (1971; Zbl 0226.58007)], H.-H. Kuo [Trans. Am. Math. Soc. 159, 57-68 (1971; Zbl 0222.28007)] and V. Goodman [Trans. Am. Math. Soc. 164, 411-426 (1972; Zbl 0227.46030)] – where the contraction of a Gaussian measure on a hypersurface was built – appeared only in the beginning of the 70s.
In conformity with the author’s affirmation, “the author seems to be the first to have succeeded in applying (infinite-dimensional) surface measures for solving concrete mathematical problems from the field of differential equations rather successfully”. The key idea is the following: these mathematical problems were “primary” and a surface measure was invented for their solving.
The scheme of this construction of surface measures follows the following steps:
– The linear space \(M^1\) of Borel measures in a Hausdorff locally convex topological vector space \(Z(\text{LCS})\), differentiable with respect to some linear and continuously enclosed subset \(Z_0\) is considered.
– The contraction (localization) of each measure on a surface of LCS is built and called a surface measure.
In general lines, this construction may be resumed in the following manner:
Given \(X\) a closed hypersurafce of \(Z\) which does not contains \(Z_0\), \(a{\i}Z_0\setminus X\), \(R_a\) a one-dimensional subspace generated by the vector \(a\), \(A\) an open subset of \(X\), open in \(X\), \(f: A\to R_a\) a continuous function having a continuous bounded derivative \(f': A\to (X\cap Z_0)^*\) along the normed space \(X\cap Z_0\), the author considers \(G= G(X, A,a,f)\) a graph in \(X\times R_a= Z\) of the function \(f\). For \(Q\in \Sigma_G\) and \(\mu\in M^1\), let \(T=T(Q)=\bigcup_{s\leq 0} \{Q- sa\}\), \(\mu_a(Q)= D_a\mu(T)\). Since the function \(D_a\mu:\Sigma_Z\to R^1\) is countably additive then \(\mu_a: \Sigma_G\to R^1\) is a Borel measure on \(G\). From the suppositions about \(f\) it follows that the function \(\omega\mapsto|(n(\omega), a)|^{- 1}\) (\(n= n(\omega)\in Z^*_0\) being the unit normal to the set \(G\) in the point \(\omega\in G\)) is bounded on \(G\), so that the measure \(\mu_G:\Sigma_G\to R^1:Q\to\int_Q|(n,a)|^{-1}d\mu_a\) is correctly defined. In this point, a main goal is to prove that the measure \(\mu_G\) is fully defined by the measure \(\mu\) and by the set \(G\), i.e., it does not depend on the coordinate representation of \(G\). Indeed, Theorem 2.2.1 of the invariance is fundamental in the surface measures construction and prove that for two sets \(G\) and \(\widetilde G\), two measures \(\mu_G\), \(\mu_{\widetilde G}\) are defined on \(G_0=G\cap\widetilde G\) and \(\mu_G=\mu_{\widetilde G}\).
Now, a set \(\Omega\in\Sigma_Z\) is called a smooth surface if for any point \(\omega\in \Omega\) there exists a neighbourhood (in \(Z\)) \(U(\omega)\) and the objects \(X\), \(A\), \(a\), \(f\) such that \(\Omega\cap U(\omega)=G\); a simple surface of \(\Omega\) is a Borel subset of a smooth surface and a surface of \(\Omega\) is a countable union of simple surface.
For the smooth surface \(\Omega\), the author defines the functions \[ (\mu^\pm)_\Omega: \Sigma_\Omega\to [0,\infty]: (\mu^\pm)_\Omega(Q)= \sup\Biggl\{\sum_m (\mu^\pm)_{G_m} (Q\cap Q_m)\Biggr\}, \] where sup is taken with respect to all finite collections of the nonintersecting sets \(Q_m\in \Sigma_{G_m}\), \(G_m= G(X_m, A_m, a_m, f_m)\). If \(\Omega\) is a surface \(\Omega= \bigcup^\infty_{m=1}\Omega_m\), \(\Omega_m\) are simple surfaces, \(\Omega_i\cap \Omega_j=\emptyset\), then the author defines \[ (\mu^\pm)_\Omega: \Sigma_\Omega\to [0,\infty]: (\mu^\pm)_\Omega(Q)= \sum^\infty_{m= 1} (\mu^\pm)_{\Omega_m}(Q\cap \Omega_m). \] The second main result of the section 2.2 is
Theorem 2.2.2. The functions \((\mu^\pm)_\Omega\) are correctly defined (i.e., they depend only on \(\Omega\), \(\mu\)), countably additive and possess the Radon property. If \(\Omega\) is a simple surface then the measures \((\mu^\pm)_\Omega\) are locally finite (and consequently Radon).
For a surface \(\Omega\), let \(\Sigma_\Omega(\mu)\) be the collection of all sets \(Q\) for \(\Sigma_\Omega\) such that \(|\mu|_\Omega(Q)< \infty\); obviously \(\Sigma_\Omega\) is a \(G\)-ring. The function \(\mu_\Omega: \Sigma_\Omega\to R^1: Q\mapsto (\mu^+)_\Omega(Q)- (\mu^-)_\Omega(Q)\) is defined. This is a surface measure on \(\Omega\).
– Finally, the third step is the following: different properties of surface measures and integrals with the respect to them and also different properties of the correspondence volume measure – surface measure are established.
Chapter 2 (sections 2.2-2.7) of the work which carried out this scheme begins with an auxiliary, but indeed absolutely necessary first section dedicated to: the definition of a measure derivative; the essential properties of differentiable measures and some results which have an independent meaning (such as the factorization theorem which permits a decomposition of a smooth measure on LCS into the product of smooth measures (one of which is transitional) in the subspaces). Also Chapter 1 may be considered which is dedicated to the integrals of vector-valued functions with respect to vector-valued measures. Some results given here, such as the theorems of passage to the limit under the sign of a vector integral and the theorems of a Fubini type, have also an independent meaning.
Related to the surface integration theory, developed in Chapter 2 of the book, the following aspect, which the author points out in the introduction, is very important: this theory is not simply a systematization and development of the author’s such a theory from the large series of papers, but an essential improvement and generalization of these results. The new variant of the theory, from Chapter 2 of the book is free of some deficiencies, the principal being the restriction of the separable Banach manifolds, the integration being possible only with respect to the surfaces being smooth in a Fréchet sense along all the directions of a basic space.
Moreover, if the initial LCS is a finite-dimensional Euclidean space and the volume measure is Lebesgue, then the corresponding surface measure coincides with the classic (geometric) measure, and the basic formulae of the theory coincide with corresponding classic measures.
The third chapter is dedicated to the applications of the theory: the distributions in an infinite-dimensional space and differential equations for them (sections 3.1-3.2), integral representation of functions on a Banach space, Green’s measure (section 3.3), the parabolic and elliptic equations in a space of measures (section 3.4), the smoothness of distributions of stochastic functionals (section 3.5), approximation of functions of an infinite-dimensional argument (section 3.6), a differentiable Urysohn function (section 3.7) and calculus of variations on a Banach space (section 3.8).
Now some words about the categories of readers which may be interested in this book (in opinion of the author and of myself): the specialists in functional analysis, measure and probability theories, the mathematicians who want to assimilate the surface integration theory as an instrument for solving some problems from other fields and the mathematicians who have no desire to know what surface integrals are. They can get acquainted with the final results of Chapter 3. (The work is presented such that the sections of Chapter 3, except the second, can be read independently.) For the specialists in infinite-dimensional differential geometry, as me, especially the section 2.8 is important; this is dedicated to the case of the Banach surfaces.
Finally, a special word about the honesty of the references. In this idea, a final section intitled Comments, the Preface and the Introduction, give an exhaustive picture of the researches in this ardent domain of the contemporary mathematics.