Negligible Sets of Radon Measures

Let m be a Radon measure on a Hausdorff topological space X. Corresponding to three kinds of outer measures, three kinds of m-negligible sets are considered. The main theorem states that in a metacompact space X each locally m-negligible set is m-negligible. For a Radon measure in a HausdorfT topological space X we distinguish three kinds of negligible sets corresponding to three kinds of outer measures: strictly negligible, negligible and locally negligible sets, respectively. It is shown that two classes of negligible sets coincide iff the corresponding outer measures coincide. Further, each locally negligible set is strictly negligible if X is Lindelöf. We give two examples that both in locally compact separable spaces and in locally compact metrizable spaces a negligible set need not be strictly negligible. It is an unsolved problem whether a locally negligible set is always negligible, or equivalently, whether the corresponding outer measures are always the same. In this latter form the problem is due to Schwartz [5, p. 17]. The main purpose of this note is to prove that in a metacompact space, and hence in a paracompact space, each locally negligible set is negligible. It will turn out that this is even true for compact inner regular Borel measures which are only locally a-finite. At least in this slightly more general situation it can be shown that the assumption " metacompact" cannot be replaced by "locally compact". By the compact inner regularity of Radon measures the necessity to distinguish several kinds of negligible sets arises with the infinity of the measures. That is, for finite, compact inner regular Borel measures the three kinds of negligible sets coincide. In the case of finite Borel measures (not necessarily compact inner regular) there are at most two kinds of negligible sets. The class of topological spaces in which these two kinds coincide for each finite or each finite regular Borel measure was considered by Gardner in [3] (see also [4]). Throughout this paper X is a Hausdorff topological space. ®(X), \}(X) and Si(X) denote, respectively, the families of all open, closed and compact subsets of X. The Borel a-algebra is denoted by %5(X). A Borel measure in X, i.e. a nonnegative, countably additive set function on 93 (X), is called locally finite if each point in X has a neighbourhood of finite measure. Received by the editors October 20, 1982. 1980 Mathematics Subject Classification. Primary 28C15. ©1983 American Mathematical Society 0002-9939/83 $1.00 + $.25 per page