Decompositions of vector measures in Riesz spaces and Banach lattices.

*(English)*Zbl 0569.28011This 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 countably 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 are compared and combined.

##### MSC:

28B15 | Set functions, measures and integrals with values in ordered spaces |

28B05 | Vector-valued set functions, measures and integrals |

46G10 | Vector-valued measures and integration |

##### Keywords:

Jordan decomposition theorem; order bounded vector measures in an order complete Riesz space; vector measures of bounded variation in a Banach space; purely finite additivity; absolute continuity; singularity; decomposition theorems of the Yosida-Hewitt and Lebesgue type; Banach lattice
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\textit{K. D. Schmidt}, Proc. Edinb. Math. Soc., II. Ser. 29, 23--39 (1986; Zbl 0569.28011)

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