Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T02:32:30.193Z Has data issue: false hasContentIssue false

Interchange of vector valued integrals when the measures are Bochner or Pettis indefinite integrals

Published online by Cambridge University Press:  17 April 2009

Andre de Korvin
Affiliation:
Department of Mathematics, Indiana State University, Terre Haute, Indiana, USA.
Charles E. Roberts Jr
Affiliation:
Department of Mathematics, Indiana State University, Terre Haute, Indiana, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Necessary and sufficient conditions for the interchange of two Bochner integrals and for the interchange of two Pettis integrals are obtained. These conditions are different from those generally required in classical Fubini theorems since they do not require the construction of the cross product measure. The proof makes use of the Vitali-Hahn-Saks Theorem. It should be noted that while Fubini theorems use the cross product measure, one of the difficulties encountered is that the product measure fails to be countable additive – this is pointed out in M. Bhaskara Rao (Indiana Univ. Math. J. 21 (1972), 847–848) and Charles Swartz (Bull. Austral. Math. Soc. 8 (1973), 359–366). Most applications require the interchange of the two integrals rather than integration with respect to the product measure.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1979

References

[1]Alò, Richard A. and Korvin, Andre de, “A one-sided Fubini theorem for Gowurin measures”, J. Math. Anal. Appl. 38 (1972), 387398.CrossRefGoogle Scholar
[2]Bandyopadhyay, U.K., “On products of vector measures”, J. Austral. Math. Soc. Ser. A 19 (1975), 9196.CrossRefGoogle Scholar
[3]Berger, Marc A. and Knight, Victor J., “A Fubini theorem for iterated stochastic integrals”, Bull. Amer. Math. Soc. 84 (1978), 159160.CrossRefGoogle Scholar
[4]Diestel, J., “Applications of weak compactness and bases to vector measures and vectorial integration”, Rev. Roumaine Math. Pures Appl. 18 (1973), 211224.Google Scholar
[5]Dinculeanu, N., Vector measures (Pergamon Press, Oxford, London, Edinburgh, New York, Toronto, Sydney, Paris, Braunschweig, 1967).CrossRefGoogle Scholar
[6]Dunford, Nelson and Schwartz, Jacob J., Linear operators, Part I (Interscience [John Wiley & Sons], New York, London, 1958).Google Scholar
[7]Gelfand, I., “Abstrakte Funktionen und lineare Operatoren”, Rec. Math. Moscou (2) 4 (1938), 235286.Google Scholar
[8]Huneycutt, James E. Jr, “Products and convolutions of vector valued set functions”, Studia Math. 41 (1972), 119129.CrossRefGoogle Scholar
[9]Pettis, B.J., “On integration in vector spaces”, Trans. Amer. Math. Soc. 44 (1938), 277304.CrossRefGoogle Scholar
[10]Rao, M. Bhaskara, “Countable additivity of a set function induced by two vector-valued measures”, Indiana Univ. Math. J. 21 (1972), 847848.CrossRefGoogle Scholar
[11]Sinclair, George Edward, “A finitely additive generalization of the Fichtenholz-Lichtenstein theorem”, Trans. Amer. Math. Soc. 193 (1974), 359374.CrossRefGoogle Scholar
[12]Swartz, Charles, “The product of vector-valued measures”, Bull. Austral. Math. Soc. 8 (1973), 359366.CrossRefGoogle Scholar