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The Orlicz-Pettis theorem for topological Riesz spaces
Author(s):
Lech
Drewnowski;
Iwo
Labuda
Journal:
Proc. Amer. Math. Soc.
126
(1998),
823-825.
MSC (1991):
Primary 40A99, 46A40
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Abstract:
A finitely additive vector measure from a  -ring to a Riesz space is countably additive (exhaustive) for all Hausdorff Lebesgue topologies on the range space, or for none of them. In particular, subseries convergent series are the same for all Hausdorff Lebesgue topologies on a Riesz space.
References:
- 1.
- C. Aliprantis and O. Burkinshaw, Locally solid Riesz spaces, Academic Press, 1978. MR 58:12271
- 2.
- -, On the structure of locally solid topologies, Canad. Math. Bull. 23 (1980), 185-191. MR 81i:46007
- 3.
- L. Drewnowski, Equivalence of Brooks-Jewett, Vitali-Hahn-Saks and Nikodym theorems, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys. 20 (1972), 725-731. MR 47:431
- 4.
- N. Dunford and J.T. Schwartz, Linear operators. Part I, Interscience, New York, 1958. MR 22:8302
- 5.
- N.J. Kalton, The Orlicz-Pettis theorem, Contemporary Mathematics 2 (1980), 91-100. MR 82g:46027
- 6.
- K. Kuratowski, Topology, vol. 1, Academic Press, New York and London, 1966. MR 36:840
- 7.
- I. Labuda, Submeasures and locally solid topologies on Riesz spaces, Math. Z. 195 (1987), 179-196. MR 88j:46009
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Additional Information:
Lech
Drewnowski
Affiliation:
Faculty of Mathematics and Computer Science, A.~ Mickiewicz University, Matejki 48/49, 60--769 Poznan, Poland
Email:
drewlech@math.amu.edu.pl
Iwo
Labuda
Affiliation:
Department of Mathematics, University of Mississippi, University, Mississippi 38677
Email:
mmlabuda@vm.cc.olemiss.edu
DOI:
10.1090/S0002-9939-98-04107-0
PII:
S 0002-9939(98)04107-0
Keywords:
Subseries convergence,
countably additive vector measure,
exhaustive vector measure,
topological Riesz space,
Lebesgue topology
Received by editor(s):
April 16, 1996
Received by editor(s) in revised form:
September 9, 1996
Additional Notes:
The paper was written while the first author held a visiting position in the Department of Mathematics, University of Mississippi, in the Spring Semester of 1996. He was also partially supported by the State Committee for Scientific Research (Poland), grant no 2 P301 003 07
Communicated by:
Dale Alspach
Copyright of article:
Copyright
1998,
American Mathematical Society
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