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
d(0,F( ·)) Î L1 (T) and liml® + ¥ \fracd(lx,F(t))l = 0d(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.