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Pettis integration in locally convex spaces. (English) Zbl 0892.46003

Here we find a very well written article on Pettis integration of functions with values in Hausdorff locally convex topological vector spaces. What the authors do is to translate some of the basic extant results about Pettis integrable functions from the Banach space domain to the always swampy domain of locally convex spaces. Being succesful, and being the proofs delicate at some points, the only thing that the reader shall miss is a surprising result holding in the locally convex setting but which is either false or meaningless in the classical Banach space setting.

MSC:

46A03 General theory of locally convex spaces
46G12 Measures and integration on abstract linear spaces
46G10 Vector-valued measures and integration
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[1] Blondia, C., Integration in locally convex spaces, Simon Stevin, 55, 81-102 (1981) · Zbl 0473.46031
[2] Chatterjee, S. D., Sur l’intégrabilité de Pettis, Math. Z., 136, 53-58 (1974) · Zbl 0264.28007 · doi:10.1007/BF01189256
[3] Edgar, G. A., Measurability in a Banach space, Indiana Univ. Math. J., 26, 663-677 (1977) · Zbl 0361.46017 · doi:10.1512/iumj.1977.26.26053
[4] Edgar, G. A., Measurability in a Banach space. II, Indiana Univ. Math. J., 28, 559-579 (1979) · Zbl 0418.46034 · doi:10.1512/iumj.1979.28.28039
[5] Edgar, G. A., On pointwise compact sets of measurable functions, Measure Theory, Oberwolfach 1981 (1982), Berlin-New York: Springer, Berlin-New York · Zbl 0486.46039
[6] Fremlin, D. H.; Talagrand, M., A decomposition theorem for additive set functions, with applications to Pettis integrals and ergodic means, Math. Z., 168, 117-142 (1979) · Zbl 0393.28005 · doi:10.1007/BF01214191
[7] Horvath, J., Topological vector spaces and distributions (1966), Reading, Mass.-London-Don Mills, Ont.: Addison-Wesley, Reading, Mass.-London-Don Mills, Ont. · Zbl 0143.15101
[8] Huff, R., Remarks on Pettis integrability, Proc. Amer. Math. Soc., 96, 402-404 (1986) · Zbl 0611.28005 · doi:10.2307/2046583
[9] Hunter, R. J.; Lloyd, John, Weakly compactly generated locally convex spaces, Math. Proc. Cambridge Philos. Soc., 82, 85-98 (1977) · Zbl 0356.46004 · doi:10.1017/S0305004100053706
[10] Khurana, S. S., Weak integration of vector-valued functions, J. Indian Math. Soc., 39, 155-166 (1975) · Zbl 0294.46034
[11] Moran, W., Measures and mappings on topological spaces, Proc. London Math. Soc., 19, 493-508 (1969) · Zbl 0186.37201 · doi:10.1112/plms/s3-19.3.493
[12] Musial, K., Vitali and Lebesgue convergence theorems for Pettis integral in locally convex spaces, Atti. Sem. Mat. Fis. Univ. Modena, 35, 159-166 (1987) · Zbl 0636.28005
[13] Pallares, A. J.; Vera, G., El recorrido de la integral indefinida de Pettis, Rev. Real Acad. Cienc. Exact. Fis. Natur. Madrid, 80, 121-131 (1986) · Zbl 0628.28013
[14] Ali, Sk. J.; Chakraborty, N. D., On Dunford and Gelfand integrals in locally convex spaces, Indian J. Pure Appl. Math., 23, 203-216 (1992) · Zbl 0762.28011
[15] Thomas, G. E. F., Integration of functions with values in locally convex Suslin spaces, Trans. Amer. Math. Soc., 212, 61-81 (1975) · Zbl 0312.28014 · doi:10.2307/1998613
[16] Varadarajan, V. S., Measures on topological spaces, Mat. Sb., 55, 97, 35-100 (1961) · Zbl 0152.04202
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