Abstract: | Let (T,τ,μ) be a finite measure space, X be a Banach space, P be a metric space and let L1(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→2X is a set-valued function such that for each fixed pεP, φ(·,p) has a measurable graph and for each fixed tεT, φ(t,·) is either upper or lower semicontinuous then the Aumann integral of φ, i.e.,∫Tφ(t,p)dμ(t)= {∫Tx(t)dμ(t):xεSφ(p)}, where Sφ(p)= {yεL1(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=Rn, and they have useful applications in general equilibrium and game theory. |