Integral Functionals,Normal Integrands and Measurable Selections.

摘要

A fundamental notion in many areas of mathematics, including optimal control, stochastic programming, and the study of partial differential equations, is that of an integral functional. By this is meant an expression of the form If(x) = integral over S of f(s,x(s))mu(DS), x is a member of X where X is a linear space of measurable functions defined on a measure space (S, A, mu) and having values in a linear space E. This paper provides a thorough treatment of the properties of such functionals in the case of E = R to the n-th power, including properties of continuity convexity and duality.