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On the jordan decomposition for vector measures
Abstract
It is shown that a vector measure μ on an algebra of sets with values in an order complete Banach lattice G is the difference of two positive vector measures if either μ is bounded and G is an order complete AM-space with unit, or μ has bounded variation and there exists a positive contractive projection G" → G. This result is a complete counterpart to the corresponding one on the regularity of a bounded linear operator.
The present paper is mainly concerned with decomposition theorems of the Jordan, Yosida-Hewitt, and Lebesgue type for vector measures of bounded variation in a Banach lattice having property ( P ). The central result is the Jordan decomposition theorem due to which these vector measures may alternately be regarded as order bounded vector measures in an order complete Riesz space or as vector measures of bounded variation in a Banach space. For both classes of vector measures, properties like countable additivity, purely finite additivity, absolute continuity, and singularity can be defined in a natural way and lead to decomposition theorems of the Yosida-Hewitt and Lebesgue type. In the Banach lattice case, these lattice theoretical and topological decomposition theorems can be compared and combined.
It is well-known that a vector-valued measure from a a-algebra into an order complete vector lattice which is countably additive with respect to order convergence has an order bounded range iff the measure has a Jordan decomposition. (This result is due to Bauer in a more general form ([2], [3]); see also [6], [9] and [10]). Since there are lattice valued measures which do not have Jordan decompositions, it follows that in general there does not exist a straightforward order analogue of the Nikodym Boundedness Theorem ([4] 1.3.1) for vector measures with values in a vector lattice. In this note we consider the problem of giving sufficient conditions on a vector lattice X so that an order version of the Nikodym Boundedness Theorem holds for X-valued measures. Our methods also allow us to obtain conditions on the vector lattice X which insure that a single X-valued vector measure which is exhaustive with respect to order convergence has order bounded
This paper gives criteria for a vector-valued Jordan decomposition theorem to hold. In particular, suppose L is an order complete vector lattice and G is a Boolean algebra. Then an additive set function μ: G-» L can be expressed as the difference of two positive additive measures if and only if μ(G) is order bounded. A sufficient condition for a countably additive set function with values in c0(r), for any set r, to be decomposed into difference of countably additive set functions is given; namely, the domain being the power set of some set.
This paper gives criteria for a vector-valued Jordan decomposition theorem to hold. In particular, suppose $L$ is an order complete vector lattice and $\mathscr{A}$ is a Boolean algebra. Then an additive set function $\mu: \mathscr{A} \rightarrow L$ can be expressed as the difference of two positive additive measures if and only if $\mu(\mathscr{A})$ 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.
- J. Diestel
Uhl jr.: Vector Measures
- J Diestel
- J. Diestel