Continuity of measurable convex and biconvex operators

Thibault, Lionel

doi:10.1090/S0002-9939-1984-0727250-X

Continuity of measurable convex and biconvex operators

Author: Lionel Thibault
Journal: Proc. Amer. Math. Soc. 90 (1984), 281-284
MSC: Primary 46A40
DOI: https://doi.org/10.1090/S0002-9939-1984-0727250-X
MathSciNet review: 727250
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Abstract: We prove that a mapping from the product of two complete metrizable vector spaces into a topological vector space which is separately universally measurable and separately convex with respect to a convex cone is continuous.

[1] J. M. Borwein, Continuity and differentiability properties of convex operators, Proc. London Math. Soc. (3) 44 (1982), no. 3, 420–444. MR 656244, https://doi.org/10.1112/plms/s3-44.3.420
[2] Jens Peter Reus Christensen, Borel structures in groups and semigroups, Math. Scand. 28 (1971), 124–128. MR 0308322, https://doi.org/10.7146/math.scand.a-11010
[3] J. P. R. Christensen, Topology and Borel structure, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1974. Descriptive topology and set theory with applications to functional analysis and measure theory; North-Holland Mathematics Studies, Vol. 10. (Notas de Matemática, No. 51). MR 0348724
[4] Pal Fischer and Zbigniew Słodkowski, Christensen zero sets and measurable convex functions, Proc. Amer. Math. Soc. 79 (1980), no. 3, 449–453. MR 567990, https://doi.org/10.1090/S0002-9939-1980-0567990-5
[5] J. Hoffmann-Jorgensen, The theory of analytic sets, Aarhus Universitet Danmark, various publication series, no. 10, 1970.
[6] Mohamed Jouak and Lionel Thibault, Equicontinuity of families of convex and concave-convex operators, Canad. J. Math. 36 (1984), no. 5, 883–898. MR 762746, https://doi.org/10.4153/CJM-1984-050-8
[7] Anthony L. Peressini, Ordered topological vector spaces, Harper & Row, Publishers, New York-London, 1967. MR 0227731
[8] Laurent Schwartz, Sur le théorème du graphe fermé, C. R. Acad. Sci. Paris Sér. A-B 263 (1966), A602–A605 (French). MR 0206676
[9] L. Thibault, Continuity of measurable convex multifunctions, Multifunctions and integrands (Catania, 1983) Lecture Notes in Math., vol. 1091, Springer, Berlin, 1984, pp. 216–224. MR 785588, https://doi.org/10.1007/BFb0098814