×

On integration in Banach spaces. XI: Integration with respect to polymeasures. (English) Zbl 0793.28006

The author continues his study of the integration of \(d\)-tuples of measurable vector-valued functions with respect to a measure (polymeasure) which has values in the space of \(d\)-multilinear operators and is separately countably additive in the strong operator topology. The author introduces two spaces of integrable functions which share many of the properties of the classical \(L^ 1\)-spaces. In order to illustrate that these function spaces are appropriate for his theory of integration, the author proves analogues of the classical Vitali’s convergence theorem, Lebesgue’s dominated convergence theorem and Fubini’s theorem for these spaces.
[Part X (1988) has been reviewed in Zbl 0688.28004; see also the following review].

MSC:

28B05 Vector-valued set functions, measures and integrals
46G10 Vector-valued measures and integration
PDF BibTeX XML Cite
Full Text: EuDML

References:

[1] Chang D. K., Rao M. M.: Bimeasures and sampling theorems for weakly harmonizable processes. Stochastic Anal. Appl. 1 (1983), 21-55. · Zbl 0511.60034
[2] Chang D. K., Rao M. M.: Bimeasures and nonstationary processes. Real and Stochastic Analysis, 7-118, Wiley Ser. Probab. Math. Statist., Wiley, New York, 1986. · Zbl 0616.60009
[3] Diestel J., Uhl J. J.: Vector measures. Amer. Math. Soc. Surveys, No. 15, Providence, 1977. · Zbl 0369.46039
[4] Diestel J.: Sequences and Series in Banach spaces. Graduate Texts in Mathematics 92, Springer-Verlag, New York-Berlin-Heidelberg-Tokyo, 1984. · Zbl 0542.46007
[5] Dobrakov I.: On integration in Banach spaces, I. Czech. Math. J. 20 (95), (1970), 511 - 536. · Zbl 0215.20103
[6] Dobrakov I.: On integration in Banach spaces, II. Czech. Math. J. 20 (95), (1970), 680-695. · Zbl 0224.46050
[7] Dobrakov I.: On integration in Banach spaces, III. Czech. Math. J. 29 (104), (1979), 478-499. · Zbl 0429.28011
[8] Dobrakov I.: On integration in Banach spaces, IV. Czech. Math. J. 30 (105), (1980), 259-279. · Zbl 0452.28006
[9] Dobrakov I.: On integration in Banach spaces,V. Czech.Math.J. 30 (105), (1980), 610-628. · Zbl 0506.28004
[10] Dobrakov I., Morales P.: On integration in Banach spacees, VI. Czech. Math. J. 35 (110), (1985), 173-187. · Zbl 0628.28007
[11] Dobrakov I.: On integration in Banach spaces, VII. Czech. Math. J.38(113),(1988),434-449. · Zbl 0674.28003
[12] Dobrakov I.: On integration in Banach spaces, VIII (Polymeasures). Czech. Math. J. 37 (112), (1987), 487-506. · Zbl 0688.28002
[13] Dobrakov I.: On integration in Banach spaces, IX (Integration with respect to polymeasures). Czech. Math. J. 38 (113), (1988), 589-601. · Zbl 0688.28003
[14] Dobrakov I.: On integration in Banach spaces, X (Integration with respect to polymeasures). Czech. Math. J. 38 (113), (1988), 713-725. · Zbl 0688.28004
[15] Dobrakov I.: Remarks on the integrability in Banach spaces. Math. Slovaca 36, 1986, 323-327. · Zbl 0635.28005
[16] Dobrakov I.: On representation oflinear operators on \(XC_{0}(T,X)\). Czech. Math. J. 21 (96), (1971), 13-30. · Zbl 0225.47018
[17] Dobrakov I.: On Lebesgue pseudonorms on \(XC_{0}(T)\). Math. Slovaca 32, 1982, 327-333. · Zbl 0525.28009
[18] Dobrakov I.: Representation ofmultilinear operators on \(XC_{0}(T_{i))}. Czech. Math. J. 39 (114), (1989),288-302.\) · Zbl 0745.46048
[19] Dobrakov I.: Representation ofmultilinear operators on \(XC_{0}(T_{i}, X_{i})\). Atti Sem. Mat.Fis. Univ. Modena
[20] Dobrakov I.: On extension of vector polymeasures. Czech. Math. J. 38 (113), (1988), 88-94. · Zbl 0688.28005
[21] Dobrakov I.: On submeasures, I. Dissertationes Math. 112, Warszawa, 1974. · Zbl 0292.28001
[22] Dobrakov I., Farková J.: On submeasures, II. Math. Slovaca 30, (1980), 65-81. · Zbl 0428.28001
[23] Jefferies B.: Radon polymeasures. Bull. Austral. Math. Soc. 32, (1985), 207-215. · Zbl 0577.28002
[24] Kakihara Y.: A note on harmonizable and V-bounded processes. J. Multivariate Anal. 16, (1985), 140-156. · Zbl 0561.60041
[25] Kakihara Y.: Some remarks on Hilbert space valued stochastic processes. Research Activities7,(1985),9-17.
[26] Kakihara Y.: Strongly and weakly harmonizable stochastic processes of H-valued random variables. J. Multivariate Anal. 18, (1986), 127-137. · Zbl 0589.60034
[27] Katsaras A. K.: Bimeasures on topological spaces. Glasnik Matematički 20 (40), (1985), 35-49. · Zbl 0587.28009
[28] Kluvánek I.: Remarks on bimeasures. Proc. Amer. Math. Soc. 81 (1981), 233 - 239. · Zbl 0456.28003
[29] Merzbach E., Zakai M.: Bimeasures and measures induced by planar stochastic integrators. J. Multivariate Anal. 19, (1986), 67-87. · Zbl 0615.60034
[30] Morse M.: Bimeasures and their integral extensions. Ann. Mat. Pura Appl. (4) 39, (1955), 345-356. · Zbl 0066.04202
[31] Morse M., Transue W.: Integral representations ofbilinear functionals. Proc. Nat. Acad. Sci. U.S.A. 35, 1949, 136-143. · Zbl 0032.20901
[32] Morse M., Transue W.: C-bimeasures A and their superior integrals A*. Rend. Circ. Mat. Palermo, (2) 4, (1955), 270-300. · Zbl 0067.28001
[33] Morse M., Transue W.: C-bimeasures A and their integral extensions. Ann. of Math. (2) 64, (1956), 480-504. · Zbl 0073.27302
[34] Morse M., Transue W.: The representation ofa bimeasure on a rectangle. Proc. Nat.Acad. Sci. U.S.A., 42, (1956), 89-95. · Zbl 0073.27301
[35] Niemi H.: On the support of a bimeasure and orthogonally scattered vector measures. Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1975), no. 2, 249-275. · Zbl 0327.28009
[36] Rao M. M.: Harmonizable processes: Structure theory. L’Einseignement math., \(II^e\) sér. 28, fasc. 3-4, 1982. · Zbl 0501.60046
[37] Thomas E.: L’intégration par rapport a une mesure de Radon vectorielle. Ann. Inst. Fourier Grenoble, 20, (1970), 55-191. · Zbl 0195.06101
[38] Ylinen K.: Fourier transforms of noncommutative analogues of vector measures and bimeasures with applications to stochastic processes. Ann. Acad. Sci. Fenn. Ser. A I, 7, (1975),355-385. · Zbl 0326.43009
[39] YIinen K.: On vector bimeasures. Annali Mat.Pura Appl. (4) 777, (1978), 115-138. · Zbl 0399.46032
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.