The measure theory on infinite-dimensional structures. The surface integration theory in general locally convex space (LCS) had been constructed. Bessides of self-dependent significance (a contribution to infinite-dimensional analysis and to direction that has been named by A. N. Kolmogorov as "Non-linear probability theory"), the theory has found the applications in various mathematical domains (distributions and differential equations, stochastic processe, approximation of functions, others) and has allowed to open the new directions: calculus of variations on LCS; infinite-dimensional Lagrange problem; potential theory in LCS. (As it turned out, these directions are closely connected with control of stochastic processes.) The construction of distributions on a Hilbert space has been suggested and developed. The quite new method for investigation of infinite-dimensional differential equations has been suggested and developed (combination "Fourier-transform surface integration"). By this method, in particular, the infinite-dimensional analog of Hormander's theorem on existance of fundamental solution of linear differential equation with constant coefficients has been proved. The generalized problem of Hilbert supports of Wiener measure was solved completely. The foundations of vector integration theory were laid; the basis propozitions were proved (a vector integral is the integral of the LCS-valued function with respect to the measure, which takes values in a vector-dual space; the domain of the definition of a function and measure is an abstract measurable space. Such integrals not were considered before, but they are important for various mathematical domains.) A number of results on LCS's theory, optimization of queueing systems, discrete problems was obtained.
Main publications:
Uglanov A. V. Integrals with respect to vector-valued measures: Theoretical problems and applications // Amer. Math. Soc. Transl., ser. 2, 1995, 163, 171–184.
Uglanov A. V. Integration on infinite-dimensional surfaces and its applications. Dordrecht: Kluwer Acad. Publ., 2000.
