×

On integration in complete bornological locally convex spaces. (English) Zbl 0926.46037

Summary: A generalization of I. Dobrakov’s integral [ ibid. 20(95), 511-536 (1970; Zbl 0215.20103)] to complete bornological locally convex spaces is given.

MSC:

46G10 Vector-valued measures and integration
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
28B05 Vector-valued set functions, measures and integrals
46A17 Bornologies and related structures; Mackey convergence, etc.

Citations:

Zbl 0215.20103
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] R. G. Bartle: A general bilinear vector integral. Studia Math. 15 (1956), 337-352. · Zbl 0070.28102
[2] M. E. Ballvé: \(L^{\alpha }\) spaces for operator valued measures. Ann. Sc. Math. Québec 14 (1990), 9-15. · Zbl 0738.46023
[3] M. E. Ballvé: On the stability of the completeness of \(L^{\alpha }\) spaces. Rend. Circ. Mat. Palermo 39 (1990), 58-64. · Zbl 0708.46033 · doi:10.1007/BF02862877
[4] M. E. Ballvé: Geometry and integration for operator valued measures. J. Math. Anal. Appl. 165 (1992), 62-70. · Zbl 0755.28004 · doi:10.1016/0022-247X(92)90068-O
[5] R. Bravo, P. Jiménez Guerra: Linear operators and vector integrals. Math. Japonica 36 (1991), 255-262. · Zbl 0746.46041
[6] J. Diestel and J. J. Uhl, Jr.: Vector Measures. Providence, Rhode Island, 1977. · Zbl 0369.46039
[7] I. Dobrakov: On integration in Banach spaces, I. Czechoslovak Math. J. 20 (1970), 511-536. · Zbl 0215.20103
[8] I. Dobrakov: On integration in Banach spaces, II. Czechoslovak Math. J. 20 (1970), 680-695. · Zbl 0224.46050
[9] I. Dobrakov: On representation of linear operators on \(C_0(T,\mathbb{X})\). Czechoslovak Math. J. 21 (1971), 13-30. · Zbl 0225.47018
[10] I. Dobrakov: On integration in Banach spaces, III. Czechoslovak Math. J. 29 ((1979), 478-499. · Zbl 0429.28011
[11] I. Dobrakov: On integration in Banach spaces, IV. Czechoslovak Math. J. 30 (1980), 259-279. · Zbl 0452.28006
[12] I. Dobrakov: On integration in Banach spaces, V. Czechoslovak Math. J. 30 (1980), 610-622. · Zbl 0506.28004
[13] I. Dobrakov: On integration in Banach spaces, VI. Czechoslovak Math. J. 35 (1985), 173-187. · Zbl 0628.28007
[14] I. Dobrakov: On integration in Banach spaces, VII. Czechoslovak Math. J. 38 (1988), 434-449. · Zbl 0674.28003
[15] C. Debieve: Integration of vector valued functions with respect to vector valued measures. Rev. Roumaine Math. Pures Appl. 26 (1981), 943-957. · Zbl 0463.46038
[16] N. Dinculeanu: Vector Measures. VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. · Zbl 0142.10502
[17] N. Dunford and J. T. Schwartz: Linear Operators. Part I: General Theory. Interscience Publishers, New York, 1958. · Zbl 0084.10402
[18] P. R. Halmos: Measure Theory. Springer-Verlag, Berlin-Heidelberg-New York, 1950. · Zbl 0040.16802
[19] J. Haluška: On the continuity of the semivariation in locally convex spaces. Math. Slovaca 43 (1993), 185-192. · Zbl 0796.46030
[20] J. Haluška: On lattices of set functions in complete bornological locally convex spaces. Simon Stevin 67 (1993), 27-48. · Zbl 0815.46004
[21] J. Haluška: On a lattice structure of operator spaces in complete bornological locally convex spaces. Tatra Mt. Math. Publ. 2 (1993), 143-147. · Zbl 0793.46003
[22] J. Haluška: On convergences of functions in complete bornological locally convex spaces. Rev. Roumaine Math. Pures Appl. 38 (1993), 327-337. · Zbl 0811.46003
[23] H. Hogbe-Nlend: Bornologies and Functional Analysis. North-Holland, Amsterdam-New York-Oxford, 1977. · Zbl 0359.46004
[24] H. Jarchow: Locally Convex Spaces. Teubner, Stuttgart, 1981. · Zbl 0466.46001
[25] H. B. Maynard: A Radon-Nikodým theorem for operator valued measures. Trans. Amer. Math. Soc. 173 (1972), 449-463. · Zbl 0263.28008 · doi:10.2307/1996285
[26] J. V. Radyno: Linear Equations and Bornology. Izd. Beloruskogo gosudarstvennogo universiteta, Minsk, 1982. · Zbl 0534.46004
[27] R. Rao Chivukula and A. S. Sastry: Product vector measures via Bartle integrals. J. Math. Anal. Appl. 96 (1983), 180-195. · Zbl 0551.28009 · doi:10.1016/0022-247X(83)90035-5
[28] S. K. Roy and N. D. Charkaborty: Integration of vector valued functions with respect to an operator valued measure. Czechoslovak Math. J. 36 (1986), 198-209. · Zbl 0611.28004
[29] C. Schwartz: Integration for the Dobrakov integral. Czechoslovak Math. J. 30 (1980), 640-646. · Zbl 0506.28005
[30] C. Schwartz: Weak Fubini theorem for the Dobrakov integral. Czechoslovak Math. J. 30 (1980), 647-654. · Zbl 0506.28006
[31] W. Smith and D. H. Tucker: Weak integral convergence theorem and operator measures. Pacific J. Math. 111 (1984), 243-256. · Zbl 0569.46021 · doi:10.2140/pjm.1984.111.243
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.