The Jordan decomposition of vector-valued measures

Abstract

This paper gives criteria for a vector-valued Jordan decomposition theorem to hold. In particular, suppose L is an order complete vector lattice and 6 is a Boolean algebra. Then an additive set function ,u: 6 -* L can be expressed as the difference of two positive additive measures if and only if ,u(d) is order bounded. A sufficient condition for a countably additive set function with values in c0(F), for any set F, to be decomposed into difference of countably additive set functions is given; namely, the domain being the power set of some set. We are concerned here with vector-valued additive set functions defined on some sort of Boolean algebra with vector-values in a (generally order complete) vector lattice. The purpose of this note is to expose conditions that insure such measures can be written as a difference of positive measures, i.e., conditions for a vector-valued Jordan decomposition theorem to hold. For this reason, a measure that can be expressed as the difference of two positive additive measures will be called decomposable. The decomposability of vector measures per se was first studied by C. E. Rickart in a 1943 Duke Mathematical Journal article where he established a Lebesgue decomposition theorem for the class of strongly bounded additive measures. This result was later re-established (although it was not realized at the time) by J. J. Uhl, Jr. [10] who also presented a Yosida-Hewitt decomposition theorem for strongly bounded measures. In [3], Diestel and Faires exhibited several decomposition theorems of the Jordan type, however, they did not give necessary and sufficient conditions for the decomposability of a vector measure. Our first result supplies these conditions, although, as is indicated, only one part of the proof is new. So, let E be a Boolean algebra, S. the vector lattice of simple functions over E endowed with the uniform norm, and L a vector lattice. If tt: C -> L is an additive set function, then the operator T : S, -> L associated with yt is defined by n n T(, aic) = ai fL(ai) Received by the editors January 24, 1975 and, in revised form, February 12, 1976. AMS (MOS) subject classifications (1970). Primary 46G10; Secondary 28A25. Copyright 6) 1977, American Mathematical Society