Extension of set functions to measures and applications to inverse limit measures

D. Mallory
1975 Canadian mathematical bulletin  
Introduction. In measure theory and probability it is often useful to be able to extend a set function g to a measure p. One situation in which such an extension arises is that of obtaining limit measures for inverse (or projective) systems of measure spaces ([1], [5] ). Since such extensions do not exist in general, conditions must be placed on g in order to guarantee the existence of a measure which is an extension of g. A condition frequently assumed for this purpose is that g can be
more » ... at g can be approximated from below on every countable family of sets in its domain by a family of sets ^ with properties very similar to those of compact sets. Specifically, *% is assumed to be such that for C l9 C 2 ,. . . e#, flLiQ=^ iff for some n, flLiQ=^ (Marczewski [3]). In order to obtain inverse limit measures one then places on the inverse system restrictions which will guarantee that the generating set function satisfies the above conditions. In this paper we show that approximation of g by families of sets satisfying a weaker condition (descending property) than Marczewski's is sufficient to guarantee extension to a measure. We then apply the result to inverse systems of measures and show that inverse limit measures exist for systems which satisfy conditions other than those required by previous workers.
doi:10.4153/cmb-1975-099-1 fatcat:okge2eofmzdt5jbxbkbgtbo7de