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Bounded common extensions of charges. (English) Zbl 0807.28002

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)].

MSC:

28A10 Real- or complex-valued set functions
28A12 Contents, measures, outer measures, capacities
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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
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