On Pettis integral and Radon measures

• Autorzy: Grzegorz Plebanek
• Języki publikacji: EN

Abstrakty

EN

Assuming the continuum hypothesis, we construct a universally weakly measurable function from [0,1] into a dual of some weakly compactly generated Banach space, which is not Pettis integrable. This (partially) solves a problem posed by Riddle, Saab and Uhl [13]. We prove two results related to Pettis integration in dual Banach spaces. We also contribute to the problem whether it is consistent that every bounded function which is weakly measurable with respect to some Radon measure is Pettis integrable.

Kategorie tematyczne

• 28C15: Set functions and measures on topological spaces (regularity of measures, etc.)
• 46G10: Vector-valued measures and integration

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1998
• Tom: 156
• Numer: 2
• Strony: 183-195

Daty

wydano

1998

otrzymano

1997-06-24

poprawiono

1997-09-22

poprawiono

1997-12-01

Twórcy

autor

Grzegorz Plebanek

• Institute of Mathematics, Polish Academy of Sciences, Kopernika 18, 51-617 Wrocław, Poland

Bibliografia

• [1] K. T. Andrews, Universal Pettis integrability, Canad. J. Math. 37 (1985), 141-159.
• [2] G. A. Edgar, Measurability in a Banach space II, Indiana Univ. Math. J. 28 (1979), 559-579.
• [3] R. Frankiewicz and G. Plebanek, On nonaccessible filters in measure algebras and functionals on $L^∞(λ)*$, Studia Math. 108 (1994), 191-200.
• [4] D. H. Fremlin, Measure-additive coverings and measurable selectors, Dissertationes Math. 260 (1987).
• [5] D. H. Fremlin, Measure algebras, in: Handbook of Boolean Algebras, J. D. Monk (ed.), North-Holand, 1989, Vol. III, Chap. 22.
• [6] D. H. Fremlin, Real-valued measurable cardinals, in: Israel Math. Conf. Proc. 6, 1993, 961-1044.
• [7] K. Kunen, Some points in βN, Math. Proc. Cambridge Philos. Soc. 80 (1975), 385-398.
• [8] K. Kunen, Set Theory, Stud. Logic 102, North-Holland, 1980.
• [9] K. Musiał, Topics in the theory of Pettis integration, Rend. Inst. Mat. Univ. Trieste 23 (1991), 177-262.
• [10] S. Negrepontis, Banach spaces and topology, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan (eds.), North-Holland, 1984, 1045-1142.
• [11] G. Plebanek, On Pettis integrals with separable range, Colloq. Math. 64 (1993), 71-78.
• [12] L. H. Riddle and E. Saab, On functions that are universally Pettis integrable, Illinois J. Math. 29 (1985), 509-531.
• [13] L. H. Riddle, E. Saab and J. J. Uhl, Sets with the weak Radon-Nikodym property in dual Banach spaces, Indiana Univ. Math. J. 32 (1983), 527-541.
• [14] G. F. Stefansson, Universal Pettis integrability, Proc. Amer. Math. Soc. 125 (1993), 1431-1435.
• [15] M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. 51 (1984).
• [16] G. Vera, Pointwise compactness and continuity of the integral, Rev. Mat. (1996) (numero supl.), 221-245.