##
**Decompositions of vector measures in Riesz spaces and Banach lattices.**
*(English)*
Zbl 0569.28011

This 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
PDFBibTeX
XMLCite

\textit{K. D. Schmidt}, Proc. Edinb. Math. Soc., II. Ser. 29, 23--39 (1986; Zbl 0569.28011)

Full Text:
DOI

### References:

[1] | DOI: 10.2307/1996758 · Zbl 0297.46034 |

[2] | Iglesias, Stochastica 5 pp 45– (1981) |

[3] | Bilyeu, Rocky Mountain J. Math. 7 pp 629– (1977) |

[4] | DOI: 10.1112/jlms/s2-3.4.672 · Zbl 0213.33901 |

[5] | DOI: 10.2307/2045357 · Zbl 0534.46017 |

[6] | DOI: 10.1007/BFb0073702 |

[7] | DOI: 10.2307/2041128 · Zbl 0339.46033 |

[8] | DOI: 10.1215/S0012-7094-43-01061-0 · Zbl 0063.06492 |

[9] | Schaefer, Banach Lattices and Positive Operators (1974) · Zbl 0296.47023 |

[10] | DOI: 10.2307/2042834 · Zbl 0374.28011 |

[11] | Niculescu, Revue Roumaine Math. Pures Appl. 21 pp 343– (1976) |

[12] | Diestel, Vector Measures (1977) |

[13] | Luxemburg, Riesz Spaces I (1971) |

[14] | DOI: 10.1007/BFb0064272 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.