Integration on Hyperspaces

Teresa Abreu[1]^, Eusébio Corbacho[2], Vaja Tarieladze[3]

tabreu@ipca.pt, corbacho@uvigo.es, tar@gw.acnet.ge

(recebido em 26 de Maio de 2008; aceite em 19 de Setembro de 2008)

 

 

Abstract. In this paper first we present elements of a theory of integration of a quasi-uniform conoid valued functions with respect to a positive measure following the doctoral thesis of the first author "Integration on Uniform Type Conoids". Then we show that it is possible to apply this theory for integration of functions with values in the hyperspace of non-empty convex subsets of a given quasi-uniform conoid. To realize such a possibility we treat the considered hyperspace as a quasi-uniform conoid in a rather natural way (as this is done in our joint work "Uniform type Hyperspaces").

Keywords: Topological conoid, quasi-uniform monoid, quasi-uniform conoid

 

 

Full text only available in PDF format.

Texto completo disponível apenas em PDF.

 

 

Bibliografia

[1]  Abreu, T. ;  Integration on Quasi-uniform Conoids, Thesis Doctoral, Vigo, (2005).          [ Links ]

[2]  Abreu, T.; Corbacho, E.; Tarieladze, V.;  Uniform Type Conoids, Communication on International Congress of Mathematicians, Madrid, Spain, (2006). 

[3]  Abreu, T.; Corbacho, E.; Tarieladze, V.;  Uniform Type Hyperspaces, Communication on Sixth Italian-Spanish Conference on General Topology and Applications. To appear in Matematica Panonnica, Italy, (2007).

[4]  Alegre C, Ferer J.; Gregory, V.,  On Hahn-Banach theorem in certain linear quasi-uniform structures, Acta Math. Hungarica, 1999, v.82, 325--330.

[5]  Berthiaume, G.,  On Quasi-Uniformities in Hyperspaces,Proc.AMS,  66, 2, (1977), 335--343. 

[6]  Castejon, A.; Corbacho, E.; Tarieladze, V.,  Metric Monoids and Integration, Preprint, Vigo, (1997). 

[7] Cao,J.; Künzi, H.;Reilly, I.;Romaguera S.;  Quasi-uniform hyperspaces of compact subsets, Topology and its Applications,  87, (1998), 117--126. 

[8]  Fletcher, P.; Lindgren, W.,  Quasi-uniform spaces, Lecture Notes in Pure ans Applied Mathematics, Marcel Dekker, Inc., New York,  77, (1982). 

[9]  Huh, W.,  Some properties of pseudonormable semilinear spaces, J. Korean Math. Soc.,  11, (1974), 77--85.

[10]  Garcia-Raffi L. M, Romaguera S.; Sanchez-Perez,E. A.,  The bicompletion of an asymmetric normed linear space, Acta Math. Hungarica, 2002, v.97, 183--191. 

[11]  Godini, G.,  On normed almost linear spaces, Math. Ann.,  279, (1988), 449--455. 

 

 

1 IPCA – Escola Superior de Gestão, Instituto Politécnico do Cávado e do Ave, Barcelos, Portugal

2 Departamento de Matematica Aplicada 1, E. T. S. E. de Telecomunicacion, Universidad de Vigo, Vigo, Spain

3 Niko Muskhelishvili Institute of Computational Mathematics, Tbilisi-0193, Georgia