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On covering systems Mallory, Donald James

Abstract

This paper is concerned with the relationships between certain covering systems useful for set differentiation and with their application to density theorems and approximate continuity. The covering systems considered are the Vitali systems (which we call V-systems), the systems introduced by Sion (which we call S-systems), and a modification of the tile systems (which we call T-systems). It is easily checked from the definitions that V-systems are S-systems, and under slight restrictions, T-systems. We show also that under certain conditions S-systems are T-systems, and that in general the converses do not hold. Density theorems and the relationships between approximate continuity and measurability of functions are discussed for these systems. In particular, we prove that for T-systems measurable functions are approximately continuous and hence a density theorem holds.

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