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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.

On sup-measurability of multifunctions with some density properties