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 p ϵ P , φ(·, p ) has a measurable graph and for each fixed t ϵ 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.