Previous |  Up |  Next

Article

Keywords:
finitely additive integration; abstract Riemann integration; absolute continuity
Summary:
Absolute continuity for functionals is studied in the context of proper and abstract Riemann integration examining the relation to absolute continuity for finitely additive measures and giving results in both directions: integrals coming from measures and measures induced by integrals. To this end, we look for relations between the corresponding integrable functions of absolutely continuous integrals and we deal with the possibility of preserving absolute continuity when extending the elemental integrals.
References:
[1] E. de Amo, I. Chiţescu and M. Díaz-Carrillo: An approximate functional Radon-Nikodym theorem. Rend. del Circ. Mat. de Palermo 48 (1999), 443–450. MR 1731446
[2] E. de Amo, I. Chiţescu and M. Díaz-Carrillo: An exact functional Radon-Nikodym theorem for Daniell Integrals. Studia Mathematica 148 (2001), 97–110. DOI 10.4064/sm148-2-1 | MR 1881255
[3] E. de Amo and M. Díaz-Carrillo: On abstract Fubini theorems for finitely additive integration. Proc. Amer. Math. Soc. 123 (1995), 2739–2744. DOI 10.1090/S0002-9939-1995-1257101-6 | MR 1257101
[4] G. Aumann: Integralerweiterungen mittels Normen. Arch. Math. 3 (1952), 441–450. DOI 10.1007/BF01900560 | MR 0054693 | Zbl 0048.03703
[5] S. Bochner: Additive set functions on groups. Ann. of Math. 40 (1939), 769–799. DOI 10.2307/1968893 | MR 0000669 | Zbl 0024.04203
[6] M. Díaz-Carrillo: Handbook of Measure Theory, Vol. 1, Chap. 11. 2002.
[7] M. Díaz-Carrillo and H. Günzler: Abstract Daniell-Loomis spaces. Bull. Austral. Math. Soc. 53 (1996), 135–142. DOI 10.1017/S0004972700016804 | MR 1371921
[8] M. Díaz-Carrillo and H. Günzler: Daniell-Loomis integrals. Rocky Mount. J. Math. 27 (1997), 1057–1087. MR 1627666
[9] M. Díaz-Carrillo and P. Muñoz-Rivas: Finitely additive integration: integral extension with local-convergence. Ann. Sci. Math. Québec 17 (1993), 145–154. MR 1259371
[10] M. Díaz-Carrillo and P. Muñoz-Rivas: Positive linear functionals and improper integration. Ann. Sci. Math. Québec 18 (1994), 149–156. MR 1311751
[11] N. Dunford and J. T. Schwartz: Linear Operartors, part I, General Theory. Interscience, New-York, 1957. MR 1009162
[12] C. Fefferman: A Radon-Nikodym theorem for finitely additive set functions. Pacific J. Math. 23 (1967), 35–45. DOI 10.2140/pjm.1967.23.35 | MR 0215956 | Zbl 0181.14801
[13] H. Günzler: Integration. Bibliographisches Institut, Mannheim, 1985. MR 0802205
[14] L. H. Loomis: Linear functionals and content. Amer. J. Math. 76 (1954), 168–182. DOI 10.2307/2372407 | MR 0060145 | Zbl 0055.10101
[15] W. F. Pfeffer: Integrals and Measures. Dekker, New-York, 1977. MR 0460580 | Zbl 0362.28004
[16] M. H. Stone: Notes on integration I–IV. Proc. Nat. Acad. Sci., USA 34 (1948), 336–342, 447–455, 483–490. DOI 10.1073/pnas.34.10.483 | MR 0025552
[17] F. W. Schäfke: Lokale Integralnormen and verallgemeinerte uneigentlich Riemann-Stiltjes-Integrals. J.Reine Angew. Math. 289 (1977), 118–134. MR 0453968
Partner of
EuDML logo