On the Upper and Lower Semicontinuity of the Aumann Integral
نویسنده
چکیده
Let (T,r,p) be a finite measure space, X be a Banach space, P be a metric space and let L,(y,X) denote the space of equivalence classes of X-valued Bochner integrable functions on (T, T, p). We show that if $I: T x P-2x is a set-valued function such that for each fixed p E P, 4(. , p) has a measurable graph and for each fixed TV T, 4(t;) is either upper or lower semicontinuous then the Aumann integral of I$, i.e., S&(t,P) d&)= {Irx(f)d~(r):xES~(p)), where S,(P) = {yEL,(p,X):y(t)E+(t,p)p-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”, and they have useful applications in general equilibrium and game theory.
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