Integration of functions with values in locally convex Suslin spaces

Thomas, G. Erik F.

doi:10.1090/S0002-9947-1975-0385067-1

Integration of functions with values in locally convex Suslin spaces
HTML articles powered by AMS MathViewer

by G. Erik F. Thomas PDF

Trans. Amer. Math. Soc. 212 (1975), 61-81 Request permission

Abstract:

The main purpose of the paper is to give some easily applicable criteria for summability of vector valued functions with respect to scalar measures. One of these is the following: If E is a quasi-complete locally convex Suslin space (e.g. a separable Banach or Fréchet space), $H \subset E’$ is any total subset, and f is an E-valued function which is Pettis summable relative to the ultra weak topology $\sigma (E,H)$. f is actually Pettis summable for the given topology. (Thus any E-valued function for which the integrals over measurable subsets can be reasonably defined as elements of E is Pettis summable.) A class of “totally summable” functions, generalising the Bochner integrable functions, is introduced. For these Fubini’s theorem, in the case of a product measure, and the differentiation theorem, in the case of Lebesgue measure, are valid. It is shown that weakly summable functions with values in the spaces $D,E,S,D’,E’,S’$, and other conuclear spaces, are ipso facto totally summable.

References

Garrett Birkhoff, Integration of functions with values in a Banach space, Trans. Amer. Math. Soc. 38 (1935), no. 2, 357–378. MR 1501815, DOI 10.1090/S0002-9947-1935-1501815-3

Integration von Funktionen, derer Werte die Elemente eines Vectorräumes sind

20

N. Bourbaki, Eléments de mathématique. XVIII. Première partie: Les structures fondamentales de l’analyse. Livre V: Espaces vectoriels topologiques. Chapitre III: Espaces d’applications linéaires continues. Chapitre IV: La dualité dans les espaces vectoriels topologiques. Chapitre V: Espaces hilbertiens, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1229, Hermann & Cie, Paris, 1955 (French). MR 0077882

Eléments de mathématique

Les structures fondamentales de l’analyse

Intégration

23

Jens Peter Reus Christensen, Borel structures and a topological zero-one law, Math. Scand. 29 (1971), 245–255 (1972). MR 313467, DOI 10.7146/math.scand.a-11050
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Lawrence M. Graves, Riemann integration and Taylor’s theorem in general analysis, Trans. Amer. Math. Soc. 29 (1927), no. 1, 163–177. MR 1501382, DOI 10.1090/S0002-9947-1927-1501382-X
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
T. H. Hildebrandt, Integration in abstract spaces, Bull. Amer. Math. Soc. 59 (1953), 111–139. MR 53191, DOI 10.1090/S0002-9904-1953-09694-X
B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), no. 2, 277–304. MR 1501970, DOI 10.1090/S0002-9947-1938-1501970-8
Albrecht Pietsch, Nukleare lokalkonvexe Räume, Schriftenreihe Inst. Math. Deutsch. Akad. Wiss. Berlin, Reihe A, Reine Mathematik, Heft 1, Akademie-Verlag, Berlin, 1965 (German). MR 0181888
Laurent Schwartz, Radon measures on arbitrary topological spaces and cylindrical measures, Tata Institute of Fundamental Research Studies in Mathematics, No. 6, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1973. MR 0426084

The Lebesgue-Nikodym theorem for vector valued Radon measures