The paper is concerned with sup-measurability of a multifunction F defined on the product X × Y of metric spaces with some differentiation bases. We introduce the lower D α property and the upper D α property of multifunction, where α ∈ (0, 1) ⊂ ℝ, and we prove sup-measurabilty of F when it has the upper D α property at ( x, y ), and F ( x , ·) has the lower D α property at y for every ( x, y ) ∈ X × Y . Some application of this theorem to the existence of solutions of differential inclusions f ′ ( x ) ∈ F ( x, f ( x )) is given.
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