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 $\mathcal {A}$ is a Boolean algebra. Then an additive set function $\mu :\mathcal {A} \to L$ can be expressed as the difference of two positive additive measures if and only if $\mu (\mathcal {A})$ is order bounded. A sufficient condition for a countably additive set function with values in ${c_0}(\Gamma )$, for any set $\Gamma$, to be decomposed into difference of countably additive set functions is given; namely, the domain being the power set of some set.