Let \({\mathcal R}\) be a ring of subsets of a set S and let G be a commutative Hausdorff topological group. An additive set function \(\mu:{\mathcal R}\to G\) is called strongly bounded (s-bounded for short) or exhaustive provided that \(\lim_{n}\mu (X_ n)=0\) for every sequence \((X_ n)_{n\in N}\) of pairwise disjoint elements of \({\mathcal R}\). In the special case where G is a normed linear space, this notion was introduced, under the first of those names, by C. E. Rickart in the forties. In that setting he proved that s-boundedness implies boundedness of the range of \(\mu\) and the converse holds if G is finite-dimensional but not in general. Moreover, he noted that, in case \({\mathcal R}\) is a \(\sigma\)-ring, s-boundedness is implied by \(\sigma\)-additivity of \(\mu\). Rickart’s main concern was to establish some Lebesgue type decomposition theorems for s-bounded set functions, thereby generalizing the corresponding results for (real) measures.
General G-valued s-bounded set functions and measures have been studied since the mid-sixties. The study reached a climax in the years 1971-75 with the appearance of some 50 papers by numerous authors and a book by M. Sion [”A theory of semigroup valued measures” (1973; Zbl 0312.28016)]. As a result, a theory emerged which generalizes and often improves various elements of both the classical measure theory and that of Banach-space-valued measures. This theory has also important connections with (nonlocally convex) topological linear spaces and with (topological) Boolean algebras. The book by Sion mentioned above and a recent monograph by C. Constantinescu [”Spaces of measures” (1984; Zbl 0555.28004)] cover only a relatively small part of the theory of group-valued set functions. The book under review is a third one, not counting a booklet by V. N. Aleksyuk [”Set functions” (1982; Zbl 0565.28002)], devoted to this theory, and as such is a welcome addition to the literature.
The author concentrates on s-bounded group-valued set functions and pays relatively little attention to measures. The book is largely self- contained and the presentation is, as a rule, very detailed. The material is divided into six chapters: I - Topological groups; II - Strongly bounded functions. Outer measures; III - Continuous functions and singular functions; IV - Uniformly s-bounded functions; V - Decomposition theorems; VI - Continuous functions and atomic functions.
Chapter I is of a preliminary nature and is concerned with some basic notions and results from the general theory of commutative topological groups. In particular, Cauchy sequences and nets and the corresponding notion of completeness are discussed. There is also a brief presentation of summable families. Moreover, the important theorem that a group topology is induced by a family of pseudonorms is given with a proof.
Chapter II presents a generalization, to the group setting, of the boundedness result of Rickart mentioned above. Of main concern here is, however, Sion’s method of generating an outer measure by means of an s- bounded set function \(\mu\) on the ring \({\mathcal R}\) taking values in a complete subset of G. In case \(\mu\) is, moreover, \(\sigma\)-additive, this method yields, via a generalization of the Carathéodory process, an extension of \(\mu\) to a measure on the \(\sigma\)-ring generated by \({\mathcal R}\) (M. Takahashi, Sion). The chapter concludes with a result of the reviewer to the effect that, under the same completeness assumption, \(\mu\) extends to an s-bounded set function on \({\mathcal P}(S)\) provided \(S\in {\mathcal R}\). (The reviewer [Colloq. Math. 51, 213-219 (1986)] has recently generalized his result and, to some extent, simplified its proof.)
Chapter III introduces the crucial notion of Fréchet-Nikodým topology (FN-topology for short). Such is called a group topology on \({\mathcal R}\), where the group operation is the symmetric difference of sets, with the property that the family of functions \(X\to X\cap T\), where \(T\in {\mathcal R}\), is uniformly continuous on \({\mathcal R}\). An additive set function \(\mu:{\mathcal R}\to G\) generates an FN-topology \(\Gamma_{\mu}\) on \({\mathcal R}\) and a converse also holds. It is shown that this correspondence leads to a reformulation, in the language of FN-topologies, of the notions of \(\mu\)-continuity and \(\mu\)-singularity and of the properties of \(\sigma\)- additivity and pure finite additivity of \(\mu\).
Chapter IV starts with some basic results on submeasures due to L. Drewnowski. One of them is the theorem that an FN-topology is induced by a family of submeasures. (The proof in the book, p. 107, contains a false claim that \(\Gamma_{\eta_ d}={\mathcal C}_ d.)\) Another one is a lemma on the existence of an order continuous restriction of an s-bounded submeasure defined on a \(\sigma\)-ring. Of main concern in this chapter are, however, sequences of s-bounded set functions defined on a \(\sigma\)- ring of sets. Namely, a criterion for uniform s-boundedness, due to F. Cafiero (the \(\sigma\)-additive case, \(G={\mathbb{R}})\) and A. B. d’Andrea de Lucia and the author, is given. Moreover, generalizations of the classical convergence theorems due to Nikodým and Vitali, Hahn and Saks are discussed (T. Andô, J. K. Brooks and R. S. Jewett, R. B. Darst, L. Drewnowski, D. Landers and L. Rogge, I. Labuda and others).
Chapters V and VI deal with decompositions of an s-bounded set function \(\mu\) on \({\mathcal R}\) taking values in a complete subset C of G. The main result here is the following theorem due to T. Traynor, which is a common generalization of previous results of his, Darst and Drewnowski: Given an FN-topology \(\Gamma\) on \({\mathcal R}\), there exist uniquely determined s- bounded set functions \(\mu_ 1\) and \(\mu_ 2\) on \({\mathcal R}\) taking values in C such that: (i) \(\mu =\mu_ 1+\mu_ 2\); (ii) \(\mu_ 1\) is \(\Gamma\)-continuous and \(\mu_ 2\) is \(\Gamma\)-singular. Moreover, \(\mu_ 1\) and \(\mu_ 2\) are \(\mu\)-continuous. This result readily yields group-valued versions of the Lebesgue-Darst and Hewitt-Yosida decomposition theorems. It is also applied by the author to derive a generalization of the Hammer-Sobczyk theorem concerning the existence of a decomposition of \(\mu\) into its atomic and continuous parts (de Lucia and de Lucia, H. Weber).
In sum, the book contains a good deal of interesting material and should prove useful to the reader who has some experience in abstract measure theory. Its usefulness is, however, diminished by numerous drawbacks and flaws, some of which are listed below.
{Reviewer’s remarks: (1) The book contains no index of terms or symbols. Another disadvantage of the same nature is the total absence of running heads. (2) The main text is accompanied by nearly one hundred explanatory notes, which are placed at the end of each chapter. Such a composition impedes the reading a great deal. (3) As the author points out in the introduction, no discussion of the Nikodým boundedness theorem is included. Here are some other important omissions: Orlicz-Pettis type theorems, Maharam’s problem, properties of the range of an s-bounded set function, inner and outer regularity, integration theory. (4) The bibliography of the book is very incomplete. For example, the Soviet literature on the subject is totally neglected. In particular, the names of V. N. Aleksyuk, M. P. Kats, V. M. Klimkin, S. A. Malyugin, G. V. Nedogibchenko, E. A. Pecherskij, L. Y. Savel’ev and A. N. Sashenkov are not mentioned. (5) Too little attention is paid to relating the abstract material of the book with the theory of scalar- and vector-valued measures. (A glaring instance of this is the treatment of the Hewitt- Yosida theorem.) Besides, the text lacks motivating examples. In fact, concrete set functions with values in a ”genuine” topological group (such as \(\Pi)\) or a nonlocally convex topological linear space appear nowhere in the book. (6) The author uses an inadequate notion of boundedness in topological groups. As a result, the boundedness of one-point sets has to be additionally postulated (Section II.2). (7) Osservazione IV on p. 36 is misleading in three different respects. (8) The proof of Proposition VI.1.4 is based on the existence of a measurable cardinal. The assertion is, however, known to hold without that assumption (see, e.g., reference item [46] of the book, (7.1)). (9) There are slips in the proofs of Theorem I.4.3 and Lemma IV.2.2 (formula (5)) and in note \((^{20})\), p. 182, and more serious inaccuracies in the proofs of Theorems III.1.2, IV.1.4, 3.3 and 5.2. (10) The reviewer has found over 50 misprints. A few of them are nasty, e.g., ”\(V_{n-1}-V_ n\)” instead of ”\(V_{n- 1}\setminus V_ n\)” on p. 12 and the omission of a condition in note \((^ 9)\), p. 181. (11) Reference item [23] by E. Guariglia has appeared [Matematiche 37(1986), 328-342 (1982; Zbl 0598.28016)].}