On the extremality of regular extensions of contents and measures

Wolfgang Adamski

Commentationes Mathematicae Universitatis Carolinae (1995)

  • Volume: 36, Issue: 2, page 213-218
  • ISSN: 0010-2628

Abstract

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Let 𝒜 be an algebra and 𝒦 a lattice of subsets of a set X . We show that every content on 𝒜 that can be approximated by 𝒦 in the sense of Marczewski has an extremal extension to a 𝒦 -regular content on the algebra generated by 𝒜 and 𝒦 . Under an additional assumption, we can also prove the existence of extremal regular measure extensions.

How to cite

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Adamski, Wolfgang. "On the extremality of regular extensions of contents and measures." Commentationes Mathematicae Universitatis Carolinae 36.2 (1995): 213-218. <http://eudml.org/doc/247718>.

@article{Adamski1995,
abstract = {Let $\mathcal \{A\}$ be an algebra and $\mathcal \{K\}$ a lattice of subsets of a set $X$. We show that every content on $\mathcal \{A\}$ that can be approximated by $\mathcal \{K\}$ in the sense of Marczewski has an extremal extension to a $\mathcal \{K\}$-regular content on the algebra generated by $\mathcal \{A\}$ and $\mathcal \{K\}$. Under an additional assumption, we can also prove the existence of extremal regular measure extensions.},
author = {Adamski, Wolfgang},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {regular content; lattice; semicompact; sequentially dominated; regular content; semicompact; sequentially dominated; lattice; extremal regular measure extensions},
language = {eng},
number = {2},
pages = {213-218},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the extremality of regular extensions of contents and measures},
url = {http://eudml.org/doc/247718},
volume = {36},
year = {1995},
}

TY - JOUR
AU - Adamski, Wolfgang
TI - On the extremality of regular extensions of contents and measures
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1995
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 2
SP - 213
EP - 218
AB - Let $\mathcal {A}$ be an algebra and $\mathcal {K}$ a lattice of subsets of a set $X$. We show that every content on $\mathcal {A}$ that can be approximated by $\mathcal {K}$ in the sense of Marczewski has an extremal extension to a $\mathcal {K}$-regular content on the algebra generated by $\mathcal {A}$ and $\mathcal {K}$. Under an additional assumption, we can also prove the existence of extremal regular measure extensions.
LA - eng
KW - regular content; lattice; semicompact; sequentially dominated; regular content; semicompact; sequentially dominated; lattice; extremal regular measure extensions
UR - http://eudml.org/doc/247718
ER -

References

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  1. Adamski W., On regular extensions of contents and measures, J. Math. Anal. Appl. 127 (1987), 211-225. (1987) Zbl0644.28002MR0904223
  2. Adamski W., On extremal extensions of regular contents and measures, Proc. Amer. Math. Soc. 121 (1994), 1159-1164. (1994) Zbl0817.28002MR1204367
  3. Bierlein D., Stich W.J.A., On the extremality of measure extensions, Manuscripta Math. 63 (1989), 89-97. (1989) Zbl0663.28004MR0975471
  4. Hackenbroch W., Measures admitting extremal extensions, Arch. Math. 49 (1987), 257-266. (1987) Zbl0612.28002MR0906740
  5. Lipecki Z., Components in vector lattices and extreme extensions of quasi-measures and measures, Glasgow Math. J. 35 (1993), 153-162. (1993) Zbl0786.28002MR1220557
  6. Los J., Marczewski E., Extensions of measure, Fund. Math. 36 (1949), 267-276. (1949) Zbl0039.05202MR0035327
  7. Marczewski E., On compact measures, Fund. Math. 40 (1953), 113-124. (1953) Zbl0052.04902MR0059994
  8. Pfanzagl J., Pierlo W., Compact systems of sets, Lecture Notes in Math., Vol. 16, SpringerVerlag, 1966. Zbl0161.36604MR0216529
  9. Plachky D., Extremal and monogenic additive set functions, Proc. Amer. Math. Soc. 54 (1976), 193-196. (1976) Zbl0285.28005MR0419711
  10. Schwartz L., Radon measures on arbitrary topological spaces and cylindrical measures, Oxford UP, 1973. Zbl0298.28001MR0426084
  11. von Weizsäcker H., Remark on extremal measure extensions, Lecture Notes in Math., Vol. 794, Springer-Verlag, 1980. MR0577962

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