On the upper and lower semicontinuity of the Aumann integral

Abstract

Let ( T,τ,μ) be a finite measure space, X be a Banach space, P be a metric space and let L 1(μ, X) denote the space of equivalence classes of X-valued Bochner integrable functions on ( T,τ,μ). We show that if φ: T× P→2 X is a set-valued function such that for each fixed pϵ P, φ(·, p) has a measurable graph and for each fixed tϵ T, φ( t,·) is either upper or lower semicontinuous then the Aumann integral of φ, i.e.,∫ T φ( t, p)d μ( t)= {∫ T x( t)d μ( t): xϵS φ ( p)}, where S φ ( p)= { yϵL 1( μ, X): y( t) ϵφ( t, p) μ−a.e.}, is either upper or lower semicontinuous in the variable p as well. Our results generalize those of Aumann (1965, 1976) who has considered the above problem for X= R n , and they have useful applications in general equilibrium and game theory.