A note on comparison between Birkhoff and McShane-type integrals for multifunctions

Real Analysis Exchange

Volumn 37, Issue 2, 2012, Pages 315-324

  Boccuto, Antonio ^a   Sambucini, Anna Rita ^a  

^a UNIVERSITY OF PERUGIA   (Italy)

Author keywords

Aumann integral; Banach spaces; Birkhoff integral; McShane integral; Multifunctions; R dstrom embedding theorem

Indexed keywords

EID: 84877646672     PISSN: 01471937     EISSN: 19301219     Source Type: Journal    
DOI: 10.14321/realanalexch.37.2.0315     Document Type: Article

References (31)

1. R. J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1-112.
2. Z. Artstein and J. A. Burns, Integration of compact set-valued functions, Pacific J. Math. 58 (1975), 297-2307.
3. E. J. Balder and A. R. Sambucini, A note on strong convergence for Pettis integrable function, Vietnam J. Math. 31(3) (2003), 341-3347.
4. E. J. Balder and A. R. Sambucini, On weak compactness and lower closure results for Pettis integrable (multi)functions, Bull. Pol. Acad. Sci. Math. 52(1) (2004), 53-561.
5. A. Boccuto and A. R. Sambucin, A McShane integral for multifunctions, J. Concr. Appl. Math. 2(4) (2004), 307-3325.
6. C. L. Byrne, Remarks on the Set-Valued Integrals of Debreu and Aumannn, J. Math. Anal. Appl. 62 (1978), 243-2246.
7. G. Birkhoff, Integration of functions with values in a Banach space, Trans. Amer. Math. Soc. 38 (1935), 357-3378.
8. B. Cascales, V. Kadets and J. Rodríguez, The Pettis integral for multi-valued functions via single-valued functions, J. Math. Anal. Appl. 332 (2007), 1-110.
9. B. Cascales, V. Kadets and J. Rodríguez, Measurable selectors and set-values Pettis integral in non-separable Banach spaces, J. Funct. Anal. 256 (2009), 673-6699.
10. B. Cascales and J. Rodríguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), 540-5-60.
11. B. Cascales and J. Rodríguez, The Birkhoff integral and the property of Bourgain, Math. Ann. 331 (2005), 259-2279.
12. C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag (1977).
13. L. Di Piazza and K. Musiał, Set-Valued Kurzweil-Henstock-Pettis Integral, Set-Valued Anal. 13 (2005), 167-1179.
14. L. Di Piazza and K. Musiał, A Decomposition Theorem for Compact-Valued Henstock Integral, Monatsh. Math. 148 (2006), 119-1126.
15. K. El Amri and C. Hess, On the Pettis integral of closed valued multi-functions, Set-Valued Anal. 8 (2000), 329-3360.
16. D. H. Fremlin, Measurable functions and almost continuos functions, Manuscripta Math. 33 (1981), 387-3405.
17. D. H. Fremlin, Integration of vector-valued functions, Atti Semin. Mat. Fis. Univ. Modena 42 (1994), 205-2211.
18. D. H. Fremlin, The generalized McShane integral, Illinois J. Math. 39 (1995), 39-367.
19. D. H. Fremlin, The McShane and Birkhoff integrals of vector-valued functions, University of Essex, Mathematics Department Research Report 92-10, version 18.5.2007, available at URL http://www.essex.ac.uk/maths/people/fremlin/preprints.htm
20. D. H. Fremlin, Measure Theory volume four: topological measure spaces Torres Fremlin, 25 Ireton Road, Colchester CO33AT, England (2003).
21. D. H. Fremlin and J. Mendoza, On the integration of vector-valued functions, Illinois J. Math. 38 (1994), 127-1147.
22. R. Gordon, The McShane Integral of Banach valued functions, Illinois J. Math. 34 (1990), 557-5567.
23. A. Martellotti and A. R. Sambucini, On the comparison between Aumann and Bochner integrals, J. Math. Anal. Appl. 260(1) (2001), 6-617.
24. A. Martellotti and A. R. Sambucini, The finitely additive integral of multifunctions with closed and convex values, Z. Anal. Anwend. 21(4) (2002), 851-8864.
25. E. J. McShane, A Riemann-type integral that includes Lebesgue-Stieltjes, Bochner and stochastic integrals. Mem. Amer. Math. Soc. 88, Amer. Math. Soc., Providence, R. I. (1969).
26. K. Musiał, Topics in the theory of Pettis integration, Rend. Istit. Mat. Univ. Trieste, 23 (1991), 177-1262.
27. B. J. Pettis, On integration in vector spaces. Trans. Amer. Math. Soc., 44(2) (1938), 277-2304.
28. H. Rådstrom, An Embedding Theorem for Spaces of Convex Sets, Proc. Amer. Math. Soc. 3 (1952), 165-1169.
29. A. R. Sambucini, Remarks on set valued integrals of multifunctions with non empty, bounded, closed and convex values, Comment. Math. Prace Mat. 39 (1999), 153-1165.
30. A. R. Sambucini, A survey on multivalued integration, Atti Semin. Mat. Fis. Univ. Modena 50 (2002), 53-563.
31. M. Talagrand, Pettis integral and measure theory. Mem. Amer. Math. Soc. 307, Am. Math. Soc., Providence, R. I. (1984).