REGULARITY FOR SET MULTIFUNCTIONS IN HAUSDORFF TOPOLOGY

As it is well-known, regularity is an important property of continuity, which connects measure theory and topology, aproximating general Borel sets by more tractable sets, such as compact and/or open sets. In this paper we study different types of regularity for monotone uniformly au-tocontinuous set multifunctions µ defined on a ring of subsets of a locally compact, Hausdorff space T and taking values in Pf (X), the family of closed, nonvoid sets of a real normed space X, family which is endowed with the Hausdorff pseudometric. Particularly, if the monotone uniformly autocontinuous set multifunction is a mul-tisubmeasure, we establish relationships with other continuity properties such as exhaustivity and order continuity. Extensions of multisubmeasures by preserving the regularity are obtained. Also, if f is a µ̃-totally measurable real valued set func-tion, where µ is a monotone exhaustive uniformly autocontinuous set multifunction, a Lusin type theorem is discussed. Theorem 1. i) µ is R-regular if and only if it is Rl-regular and Rr-regular. ii) µ is R′-regular if and only if it is R′l-regular and