On the Extension of Borel Measures

The class of Borel sets in a locally compact space is the cr-ring generated by the compact sets [3, p. 219], while the class of weakly Borel sets is the (r-algebra generated by the closed sets [2]. The object of this note is to show that any measure defined on the class of Borel sets may be extended to the class of weakly Borel sets in a simple and canonical way (Theorem 1). There is a useful application to regular Borel measures (Theorem 2) and, in particular, to Haar measure (Theorem 3). The first part of our discussion is valid for an arbitrary measure space (X, S, p) in the sense of Halmos [3, p. 73]. Recall that the sets £ in S are called measurable. We say that a subset A of X is locally measurable (with respect to S) if its intersection with every measurable set is measurable, that is, if Er\AQS for every E in S. We write S\ for the class of all locally measurable sets ; it is easy to see that Sx is a (r-algebra containing S. Moreover, if Sx is regarded as a ring in the usual way, then S is an ideal in SxThe measure p may be extended to a measure p\ on Sx in the following way. For each measurable set E, let pE be the measure on Sx defined by the formula pE(A) =p(EP\A); the family of measures {pE:EQ§>} is increasingly directed in the obvious sense, and, defining