Jordan Decompositions of Generalized Vector Measures

This ``research note'' looks at the Jordan decomposition of vector measures from a Boolean ring into a Riesz space or Banach lattice. The treatment is relatively self-contained. The author uses a common approach to the Jordan decomposition of vector measures and linear operators, allowing a smoother application of the general results to order-bounded vector measures and linear operators in a Riesz space. It also presents the first unified treatment of the Jordan decomposition on norm-bounded vector measures and linear operators in a Banach lattice. Incidentally, these general results are also applicable to the problem of developing a satisfactory measure theory on systems of fuzzy sets. In general, this research note demonstrates that additive functions on a commutative clan provide a useful tool for unifying and extending parts of measure and operator theory.