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**Jordan decompositions of generalized vector measures.**
*(English)*
Zbl 0692.28004

Pitman Research Notes in Mathematics Series, 214. Harlow: Longman Scientific & Technical; New York etc.: John Wiley & Sons, Inc. 142 p. £15.00 (1989).

In this book the author gives an extensive and fundamental survey of Jordan decomposition-type theorems and offers several illuminating applications to vector-valued measures on a Boolean ring or to linear operators on a normed Riesz space.

The book consists of six chapters. In Chapter 1 the author introduces the reader in a new concept of the commutative minimal clan, a common abstraction of Boolean rings and commutative lattice-ordered groups. More precisely, a minimal clan is a set E equipped with a relation \(S\subseteq E\times E\) (the set of summable elements of E), a partial addition \(+:S\to E,\) is and an order relation \(\leq\) such that \((E,S,+,\leq)\) is a commutative lattice-ordered partial semigroup having the cancellation property and the difference property which is closely related to the modular law in commutative lattice-ordered groups. The axioms imply that the lattice operations are distributive. A commutative minimal clan is a Boolean ring if and only if no other elements are summable and it is a commutative lattice-ordered group if and only if each element is invertible. Jordan decomposition-type theorems of additive set functions from a commutative minimal clan E into a Riesz space or normed lattice G are discussed in Chapter 2. As it is well known the first Jordan decompositions of vector measures were obtained by H. Bauer [Sitzungsber. Bayer. Akad. Wiss., Math. Naturwiss. Kl. 1953, 89-117 (1953; Zbl 0052.265); J. Reine Angew. Math. 194, 141-179 (1955; Zbl 0065.017)] as a consequence of his general results on normalized valuations of relatively bounded variation from a distributive lattice into an order complete Riesz space. On the other hand Chapters 2, 3 deal with Riesz (resp. normed lattices) from a commutative minimal clan E into a Riesz space (resp. Banach lattice) G. Furthermore the author discusses in Chapter 4 band decompositions of order bounded additive functions from E into G. Nice applications are provided in Chapters 5, 6 of previous general results on additive functions on E to vector (resp. partially ordered semigroup)-valued measures or to linear operators on G.

In the reviewer’s opinion the author has done a marvellous work in bringing together Jordan decomposition-type theorems of vector-valued measures and normed lattice-valued linear operators.

The book consists of six chapters. In Chapter 1 the author introduces the reader in a new concept of the commutative minimal clan, a common abstraction of Boolean rings and commutative lattice-ordered groups. More precisely, a minimal clan is a set E equipped with a relation \(S\subseteq E\times E\) (the set of summable elements of E), a partial addition \(+:S\to E,\) is and an order relation \(\leq\) such that \((E,S,+,\leq)\) is a commutative lattice-ordered partial semigroup having the cancellation property and the difference property which is closely related to the modular law in commutative lattice-ordered groups. The axioms imply that the lattice operations are distributive. A commutative minimal clan is a Boolean ring if and only if no other elements are summable and it is a commutative lattice-ordered group if and only if each element is invertible. Jordan decomposition-type theorems of additive set functions from a commutative minimal clan E into a Riesz space or normed lattice G are discussed in Chapter 2. As it is well known the first Jordan decompositions of vector measures were obtained by H. Bauer [Sitzungsber. Bayer. Akad. Wiss., Math. Naturwiss. Kl. 1953, 89-117 (1953; Zbl 0052.265); J. Reine Angew. Math. 194, 141-179 (1955; Zbl 0065.017)] as a consequence of his general results on normalized valuations of relatively bounded variation from a distributive lattice into an order complete Riesz space. On the other hand Chapters 2, 3 deal with Riesz (resp. normed lattices) from a commutative minimal clan E into a Riesz space (resp. Banach lattice) G. Furthermore the author discusses in Chapter 4 band decompositions of order bounded additive functions from E into G. Nice applications are provided in Chapters 5, 6 of previous general results on additive functions on E to vector (resp. partially ordered semigroup)-valued measures or to linear operators on G.

In the reviewer’s opinion the author has done a marvellous work in bringing together Jordan decomposition-type theorems of vector-valued measures and normed lattice-valued linear operators.

Reviewer: P.K.Pavlakos

### MSC:

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

28B10 | Group- or semigroup-valued set functions, measures and integrals |

28-02 | Research exposition (monographs, survey articles) pertaining to measure and integration |