On generation of measurable covers for measurable sets using multiple integral of functions

Abstract

A fundamental notion in many areas of mathematics, including optimization, probability, variational problems, functional analysis and operator theory, is that of an integral functional. By this is meant an expression of the form If(x) = ~ f(s,x(s))p(ds), x E X, where X is a linear space of measurable functions defined on a measure space (S,A,~) and having values in a linear space E. The function f: S x E § R is the associated Inte~rand. Classically, only finite integrands on S x R n were studied, usually under the assumption that f(s,x) was continuous in x and measurable in s (the Carathgodory condition). However, from the modern point of view it is essential to admit possibly infinite values for f and If, since it is in this way that important kinds of constraints can most efficiently be represented. Such integrands require a distinctly new theoretical approach, where questions of measurability and the existence of measurable selections are prominent and are reflected in a concept of "normality". The purpose of these notes is to provide a relatively thorough treatment of the most common case in applications, that where E = R n. While many of the results have extensions in one way or another beyond this case, as indicated to some extent in the text, these are often more complicated technically and may require further restrictions. For example, it is only for R n that one presently knows how to develop a complete theory without assuming that the measurable space is complete, an assumption which appears to be awkward in some situations. In treating Inflnite-dimenslonal spaces E, there are the usual problems of the multiplicity of topologies and dualities which must be ironed out. It is desirable, therefore, to have available a full and consistent exposition of the details in the basic case of E = R n, freeing one from the need to search for auxilliary results through sequences of papers wlth varying frameworks.