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Abstract. We consider a multifunction F : T × X → 2 E , where T , X and E are separable metric spaces, with E complete. Assuming that F is jointly measurable in the product and a.e. lower semicontinuous in the second variable, we establish the existence of a selection for F which is measurable with respect to the first variable and a.e. continuous with respect to the second one. Our result is in the spirit of [11], where multifunctions of only one variable are considered. Introduction.If X is a topological space, we denote by B(X) the Borel σ-algebra of X. Moreover, if µ is measure on B(X), we denote by T µ (X) the completion of B(X) with respect to µ. We briefly put T µ = T µ (X) when ambiguities do not occur. For the basic definitions about multifunctions, we refer the reader to [6] and [7]. This note is motivated by the main result of [11], which concerns the existence of Riemann-measurable selections (i.e., selections which are a.e. continuous) for a given multifunction. For the reader's convenience, we now state the main result of [11] (as usual, by a Polish space we mean a complete separable metric space).