Banach Journal of Mathematical Analysis

The general Fubini theorem in complete bornological locally convex spaces

Jan Haluska and Ondrej Hutnik

Full-text: Open access

Abstract

The Fubini theorem for the generalized Dobrakov integral in complete bornological locally convex topological vector spaces is proven.

Article information

Source
Banach J. Math. Anal., Volume 4, Number 2 (2010), 53-74.

Dates
First available in Project Euclid: 7 February 2011

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1297117241

Digital Object Identifier
doi:10.15352/bjma/1297117241

Mathematical Reviews number (MathSciNet)
MR2606482

Zentralblatt MATH identifier
1198.46037

Subjects
Primary: 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22]
Secondary: 28B05: Vector-valued set functions, measures and integrals [See also 46G10]

Keywords
Fubini theorem bilinear integral bornology locally convex topological vector space product measure Dobrakov integral

Citation

Haluska, Jan; Hutnik, Ondrej. The general Fubini theorem in complete bornological locally convex spaces. Banach J. Math. Anal. 4 (2010), no. 2, 53--74. doi:10.15352/bjma/1297117241. https://projecteuclid.org/euclid.bjma/1297117241


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References

  • N.U. Ahmed, Evolution equations determined by operator valued measures and optimal control, Nonlinear Analysis 67 (2007), 3199–3216.
  • M.E. Ballvé and P. Jiménez Guerra, Fubini theorems for bornological measures, Math. Slovaca 43 (1993), 137–148.
  • N. Dinculeanu, Vector measures, Pergamon Press, New York, 1967.
  • I. Dobrakov, On integration in Banach spaces, I., Czechoslovak Math. J. 20 (1970), 511–536.
  • I. Dobrakov, On integration in Banach spaces, II., Czechoslovak Math. J. 20 (1970), 680–695.
  • I. Dobrakov, On integration in Banach spaces, III., Czechoslovak Math. J. 29 (1979), 478–499.
  • I. Dobrakov, On integration in Banach spaces, VII., Czechoslovak Math. J. 38 (1988), 434–449.
  • M. Ducho\v n and I. Kluvánek, Inductive tensor product of vector valued measures, Mat. Čas. 17 (1967), 108–112.
  • M. Ducho\v n, On the projective tensor product of vector-valued measures II., Mat. Čas. 19 (1969), 228–234.
  • N. Dunford and J.T. Schwartz, Linear Operators. Part I: General Theory, Interscience Publisher, New York, 1958.
  • F.J. Fernandez, On the product of operator-valued measures, Czechoslovak Math. J. 40 (1990), 543 – 562.
  • P.P. Halmos, Measure Theory, Springer, New York, 1950.
  • J. Haluška, On convergences of functions in complete bornological locally convex spaces, Rev. Roumaine Math. Pures Appl. 38 (1993), 327–337.
  • J. Haluška, On lattices of set functions in complete bornological locally convex spaces Simon Stevin 67 (1993), 27–48.
  • J. Haluška, On a lattice structure of operator spaces in complete bornological locally convex spaces, Tatra Mt. Math. Publ. 2 (1993), 143–147.
  • J. Haluška, On convergences of functions in complete bornological locally convex spaces, Rev. Roumaine Math. Pures Appl. 38 (1993), 327–337.
  • J. Haluška, On integration in complete bornological locally convex spaces, Czechoslovak Math. J. 47 (1997), 205–219.
  • J. Haluška and O. Hutník, On integrable functions in complete bornological locally convex spaces (submitted).
  • J. Haluška and O. Hutník, On vector integral inequalities, Mediterr. J. Math. 6(1) (2009), 105–124.
  • J. Haluška and O. Hutník, The Fubini theorem for bornological product measures, Results Math. 54(1-2) (2009), 65–73.
  • J. Haluška and O. Hutník, On domination and bornological product measures, Mediterr. J. Math. (to appear).
  • E. Hille and R. Phillips, Functional Analysis and Semigroups, Providence, 1957.
  • H. Hogbe-Nlend, Bornologies and Functional Analysis, North-Holland, Amsterdam–New York–Oxford, 1977.
  • H. Jarchow, Locally convex spaces, Teubner, Stuttgart, 1981.
  • S.K. Mitter and S.K. Young, Integration with respect to operator-valued measures with applications to quantum estimation theory, Ann. Mat. Pura Appl. 137(1) (1984), 1-39.
  • J.V. Radyno, Linear equations and the bornology (in Russian), Izd. Bel. Gosud. Univ., Minsk, 1982.
  • R. Rao Chivukula and A.S. Sastry, Product vector measures via Bartle integrals, J. Math. Anal. Appl. 96 (1983), 180–195.
  • K.S. Ryu, The Dobrakov integral over paths, J. Chungcheong Math. Soc. 19(1) March 2006.
  • C. Swartz, Products of vector measures, Math. Čas. 24 (1974), 289–299.
  • C. Swartz, A generalization of a Theorem of Ducho\v n on products of vector measures, J. Math. Anal. Appl. 51 (1975), 621–628.
  • H. Weber, Topological Boolean Rings. Decomposition of finitely additive set functions, Pacific J. Math. 110(2) (1984), 471–495.
  • N.-Ch. Wong, The triangle of operators, topologies, bornologies (English summary) Third International Congress of Chinese Mathematicians. Part 1, 2, 395–421, AMS/IP Stud. Adv. Math. 42, pt.1, 2, Amer. Math. Soc., Providence, RI, 2008.