Basile, A.; Bhaskara Rao, K. P. S.; Shortt, R. M. Bounded common extensions of charges. (English) Zbl 0807.28002 Proc. Am. Math. Soc. 121, No. 1, 137-143 (1994). Let \(\mathcal A\) and \(\mathcal B\) be fields of subsets of a set \(X\) and let \(\mu: {\mathcal A}\to \mathbb{R}\) and \(\nu: {\mathcal B}\to \mathbb{R}\) be bounded finitely additive measures with \(\mu(C)= \nu(C)\) whenever \(C\in {\mathcal A}\cap {\mathcal B}\). Denote by \(\eta\) the common extension of \(\mu\) and \(\nu\) to \({\mathcal A}\cup {\mathcal B}\). The authors prove that there exists a bounded finitely additive extension of \(\eta\) to the algebra generated by \({\mathcal A}\cup {\mathcal B}\) if and only if the sums \(\sum^ N_{i= 0} | \eta(C_{i+1})- \eta(C_ i)|\), where \(\emptyset= C_ 0\subseteq C_ 1\subseteq\cdots\subseteq C_{N+1}= X\) is a chain in \({\mathcal A}\cup {\mathcal B}\) and \(N\geq 0\), are uniformly bounded. This answers a problem raised by the reviewer [Czech. Math. J. 36(111), 489-494 (1986; Zbl 0622.28007)]. As an application, partial solutions to the global version of the problem are given. Another partial solution to that version of the problem was previously given by K. D. Schmidt and G. Waldschaks [Measure theory, Proc. Conf., Oberwolfach/Ger. 1990, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 28, 117-124 (1992; Zbl 0766.28002)]. Reviewer: Z.Lipecki (Wrocław) Cited in 3 Documents MSC: 28A10 Real- or complex-valued set functions 28A12 Contents, measures, outer measures, capacities Keywords:field of sets; bounded charge; independence; finitely additive measures; common extension Citations:Zbl 0622.28007; Zbl 0766.28002 PDFBibTeX XMLCite \textit{A. Basile} et al., Proc. Am. Math. Soc. 121, No. 1, 137--143 (1994; Zbl 0807.28002) Full Text: DOI References: [1] K. P. S. Bhaskara Rao and M. Bhaskara Rao, Theory of charges, Pure and Applied Mathematics, vol. 109, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. A study of finitely additive measures; With a foreword by D. M. Stone. · Zbl 0516.28001 [2] Zbigniew Lipecki, On common extensions of two quasimeasures, Czechoslovak Math. J. 36(111) (1986), no. 3, 489 – 494. · Zbl 0622.28007 [3] Klaus D. Schmidt and Gerd Waldschaks, Common extensions of order bounded vector measures, Rend. Circ. Mat. Palermo (2) Suppl. 28 (1992), 117 – 124. Measure theory (Oberwolfach, 1990). · Zbl 0766.28002 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.