A Stone space approach to the existence of bounded common extensions (Q5906286)

Let \({\mathcal A}\) and \({\mathcal B}\) be fields of subsets of a nonempty set \(X\), and denote by \({\mathcal C}\) the field of subsets of \(X\) generated by \({\mathcal A}\cup{\mathcal B}\). The authors are concerned with the following two related problems: Find necessary and sufficient conditions on \({\mathcal A}\) and \({\mathcal B}\) in order that for arbitrary [for arbitrary 0-1 valued] \(\mu\in \text{ba}({\mathcal A})\) and \(\nu\in \text{ba}({\mathcal B})\) with \(\mu\mid{\mathcal A}\cap{\mathcal B}= \nu\mid{\mathcal A}\cap{\mathcal B}\) there exists \(\rho\in \text{ba}({\mathcal C})\) which is a common extension of \(\mu\) and \(\nu\). They give some answers to the former problem in the case where \({\mathcal A}\cap{\mathcal B}= \{\emptyset, X\}\) (Theorem 3.1) and to the latter problem in the general case (Theorem 2.1). The answers involve a distance function on the Stone space of \({\mathcal C}\) introduced by \textit{R. Göbel} and \textit{R. M. Shortt} [Fundam. Math. 146, No. 1, 1-20 (1994; Zbl 0851.28007)] for a related purpose and special chains (with respect to inclusion) of elements of \({\mathcal A}\cup{\mathcal B}\). Unfortunately, the paper is written carelessly and there are many slips, errors and misprints. In particular, the proofs of Theorems 2.1, (ii) \(\Rightarrow\) (i), and 3.1, (iii) \(\Rightarrow\) (i), are unnecessarily involved. In fact, both implications are easy consequences of the following straightfoward assertion: Let \(\mu_i\in \text{ba}({\mathcal A})\) and \(\nu_i\in \text{ba}({\mathcal B})\), \(i= 1,\dots, n\). If \(\rho_{2i- 1}[\rho_{2i}]\) is a common extension of \(\mu_i\) and \(\nu_i\) [\(\mu_{i+1}\) and \(\nu_i\)] to \({\mathcal C}\) for \(i= 1,\dots, n\), \([i=1,\dots, n-1]\), then \(\rho= \sum^{2n-1}_{i= 1} (-1)^{i+ 1}\rho_i\) is a common extension of \(\mu_1\) and \(\nu_n\) to \({\mathcal C}\). Moreover, Lemma 4.4 is, clearly, false for \(n\geq 3\). Fortunately, the conclusion the authors derive from it is true. Indeed, the following lemma holds (the notation follows that of the authors): If \(\underline A\in {\mathcal A}_n\), \(\underline B\in{\mathcal B}\) and \(\underline A\cap\underline B= \emptyset\), then there exists \(\underline B^*\in{\mathcal B}_n\) such that \(\underline A\cap\underline B^*= \emptyset\) and \(\underline B\subset\underline B^*\). Finally, we note that the implication (iii) \(\Rightarrow\) (iv) mentioned on pp. 444 and 451 was already established by the reviewer [Czech. Math. J. 36(111), 489-494 (1986; Zbl 0622.28007), Example 1]. Postscript. This review was originally prepared for MR and a copy of it, along with an extensive list of additional remarks, were sent to the authors. They then prepared a text ``Addenda and corrigenda to the paper: `A Stone space approach to the existence of bounded common extensions' '' which takes account of some of the reviewer's remarks.