The completeness in spaces of bounded pettis intergrable functions and in spaces of bounded functions satisfying the law of large numbers

• Autorzy: Musiał K.
• Języki publikacji: EN

Abstrakty

EN

It has been proven already by Pettis [5] that the space P(, X) of Pettis integrable functions may be non-complete when endowed with the semivariation norm of the integrals. Then Thomas [9] proved that the space is almost always non-complete. In view of the Open Mapping Theorem in such a case no complete equivalent norm can be defined on P(p,,X). The question is now whether there are interesting linear subsets of P(/A, X) where a complete norm does exist. In this paper we consider two such subspaces: the space Poo (/^, X) of scalarly bounded Pettis integrable functions and the space LLNoo(^,X) of scalarly bounded functions satisfying the strong law of large numbers. We prove that in several cases these spaces are complete.

Słowa kluczowe

EN

Pettis integral; law of large numbers; lifting

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2001
• Tom: Vol. 34, nr 2
• Strony: 339--344
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Musiał K.

• Mathematical Institute Wrocław University Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland

Bibliografia

• [1] D. H. Fremlin and M. Talagrand, A decomposition theorem for additive set functions, with applications to Pettis integrals and ergodic means, Math. Z. 168 (1979), 117-142.
• [2] A. Ionescu-Tulcea and C. Ionescu-Tulcea, Topics in the Theory of Lifting, Ergebnisse Math. Grenzgebiete, Band 48, Springer, 1969.
• [3] K. Musiał, Topics in the theory of Pettis integration, Rendiconti Ist. di Matematica dell'Università di Trieste 23 (1991), 177-262.
• [4] K. Musiał, The completeness problem in spaces of Pettis integrable functions, Quaestiones Math., accepted.
• [5] B. J. Pettis, On integration in vector spaces, TAMS 44 (1938), 277-304.
• [6] V. I. Rybakov, On Pettis integrability of Stone transformation, Mat. Zametki 60 (1996), 238-253 (in Russian).
• [7] M. Talagrand, Pettis integral and measure theory, Memoirs AMS 307 (1984).
• [8] M. Talagrand, The Glivenko-Cantelli problem, Ann. Probab. 15 (1987), 837-870.
• [9] G. E. F. Thomas, Totally Summable Functions with Values in Locally Convex Spaces, Measure Theory (Oberwolfach 1975), Lecture Notes in Math. 541 (1976), 117-131.