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Keywords:
regular content; lattice; semicompact; sequentially dominated
regular content; lattice; semicompact; sequentially dominated
Summary:
Let $\Cal A$ be an algebra and $\Cal K$ a lattice of subsets of a set $X$. We show that every content on $\Cal A$ that can be approximated by $\Cal K$ in the sense of Marczewski has an extremal extension to a $\Cal K$-regular content on the algebra generated by $\Cal A$ and $\Cal K$. Under an additional assumption, we can also prove the existence of extremal regular measure extensions.
References:
[1] Adamski W.: On regular extensions of contents and measures. J. Math. Anal. Appl. 127 (1987), 211-225. MR 0904223 | Zbl 0644.28002
[2] Adamski W.: On extremal extensions of regular contents and measures. Proc. Amer. Math. Soc. 121 (1994), 1159-1164. MR 1204367 | Zbl 0817.28002
[3] Bierlein D., Stich W.J.A.: On the extremality of measure extensions. Manuscripta Math. 63 (1989), 89-97. MR 0975471 | Zbl 0663.28004
[4] Hackenbroch W.: Measures admitting extremal extensions. Arch. Math. 49 (1987), 257-266. MR 0906740 | Zbl 0612.28002
[5] Lipecki Z.: Components in vector lattices and extreme extensions of quasi-measures and measures. Glasgow Math. J. 35 (1993), 153-162. MR 1220557 | Zbl 0786.28002
[6] Los J., Marczewski E.: Extensions of measure. Fund. Math. 36 (1949), 267-276. MR 0035327 | Zbl 0039.05202
[7] Marczewski E.: On compact measures. Fund. Math. 40 (1953), 113-124. MR 0059994 | Zbl 0052.04902
[8] Pfanzagl J., Pierlo W.: Compact systems of sets. Lecture Notes in Math., Vol. 16, SpringerVerlag, 1966. MR 0216529 | Zbl 0161.36604
[9] Plachky D.: Extremal and monogenic additive set functions. Proc. Amer. Math. Soc. 54 (1976), 193-196. MR 0419711 | Zbl 0285.28005
[10] Schwartz L.: Radon measures on arbitrary topological spaces and cylindrical measures. Oxford UP, 1973. MR 0426084 | Zbl 0298.28001
[11] von Weizsäcker H.: Remark on extremal measure extensions. Lecture Notes in Math., Vol. 794, Springer-Verlag, 1980. MR 0577962
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