On the integrable selections of certain multifunctions

The aim of this paper is to establish a result of which the following is a particular case: If F is a nonempty closed-valued measurable multifunction, from a nonatomic σ-finite measure space (T, F, μ) into a separable real Banach space E, such that $$d(0,F( cdot )) in L^1 (T) and mathop {lim }limits_{lambda to + infty } frac{{d(lambda x,F(t))}}{lambda } = 0$$ for almost every t ∈ T and for every x ∈ E, then each closed hyperplane of L ¹(T,E) contains a selection of F. Also, some consequences are indicated.