We are concerned with the notions of absolute continuity in terms of $0$-continuity, and of $(\epsilon,\delta)$-continuity, with respect to a finitely additive function taking values into a topological commutative group. A sufficient condition for their equivalence is established. The non-additive case of functions with values into Hausdorff topological spaces is also investigated.
References
P. Cavaliere, P. de Lucia, On Brooks-Jewett, Vitali-Hahn-Saks and Nikodým convergence theorems for quasi-triangular functions, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 20 (2009), no. 4, 387–396. MR2550853 10.4171/RLM/554 P. Cavaliere, P. de Lucia, On Brooks-Jewett, Vitali-Hahn-Saks and Nikodým convergence theorems for quasi-triangular functions, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 20 (2009), no. 4, 387–396. MR2550853 10.4171/RLM/554
P. Cavaliere, F. Ventriglia, On non-atomicity for non-additive functions, J. Math. Anal. Appl. (in press) – online at: http:/\!/dx.doi.org/10.1016/ j.jmaa.2014.01.074. MR3173173 10.1016/j.jmaa.2014.01.074 P. Cavaliere, F. Ventriglia, On non-atomicity for non-additive functions, J. Math. Anal. Appl. (in press) – online at: http:/\!/dx.doi.org/10.1016/ j.jmaa.2014.01.074. MR3173173 10.1016/j.jmaa.2014.01.074
L. Drewnowski, Topological rings of sets, continuous set functions, integration I, II, III, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 269–276, 277–286, 439–445. MR306432 L. Drewnowski, Topological rings of sets, continuous set functions, integration I, II, III, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 269–276, 277–286, 439–445. MR306432
H. König, The Lebesgue decomposition theorem for arbitrary contents, Positivity 10 (2006), no. 4, 779–793. MR2280650 10.1007/s11117-006-0045-2 H. König, The Lebesgue decomposition theorem for arbitrary contents, Positivity 10 (2006), no. 4, 779–793. MR2280650 10.1007/s11117-006-0045-2
Z. Lipecki, A characterization of group–valued measures satisfying the countable chain condition, Colloquium Mathematicum 31 (1974), no. 2, 231–234. MR364595 Z. Lipecki, A characterization of group–valued measures satisfying the countable chain condition, Colloquium Mathematicum 31 (1974), no. 2, 231–234. MR364595
D. Mitrea, I. Mitrea, M. Mitrea, E. Ziadé, Abstract capacitary estimates and the completeness and separability of certain classes of non-locally convex topological vector spaces, J. Funct. Anal. 262 (2012), 4766–4830. MR2913687 10.1016/j.jfa.2012.03.012 D. Mitrea, I. Mitrea, M. Mitrea, E. Ziadé, Abstract capacitary estimates and the completeness and separability of certain classes of non-locally convex topological vector spaces, J. Funct. Anal. 262 (2012), 4766–4830. MR2913687 10.1016/j.jfa.2012.03.012
C. P. Niculescu, An overview of absolute continuity and its applications, Inequalities and applications, 201–214, Internat. Ser. Numer. Math., 157, Birkhäuser, Basel, 2009. MR2758981 10.1007/978-3-7643-8773-0_19 C. P. Niculescu, An overview of absolute continuity and its applications, Inequalities and applications, 201–214, Internat. Ser. Numer. Math., 157, Birkhäuser, Basel, 2009. MR2758981 10.1007/978-3-7643-8773-0_19
T. Traynor, Absolute countinuity for group-valued measures, Canad. Math. Bull. 16 (1973), no. 4, 577–579. MR354998 10.4153/CMB-1973-094-4 T. Traynor, Absolute countinuity for group-valued measures, Canad. Math. Bull. 16 (1973), no. 4, 577–579. MR354998 10.4153/CMB-1973-094-4
T. Traynor, The Lebesgue decomposition for group-valued set functions, Trans. Amer. Math. Soc. 220 (1976), 307–319. MR419725 10.1090/S0002-9947-1976-0419725-8 T. Traynor, The Lebesgue decomposition for group-valued set functions, Trans. Amer. Math. Soc. 220 (1976), 307–319. MR419725 10.1090/S0002-9947-1976-0419725-8
H. Weber, Vergleich monotoner Ringtopologien und absolute Stetigkeit von Inhalten (German) [Comparison of monotone ring topologies and absolute continuity of contents], Comment. Math. Univ. St. Paul. 31 (1982), no. 1, 49–60. MR674580 H. Weber, Vergleich monotoner Ringtopologien und absolute Stetigkeit von Inhalten (German) [Comparison of monotone ring topologies and absolute continuity of contents], Comment. Math. Univ. St. Paul. 31 (1982), no. 1, 49–60. MR674580
H. Weber, FN-Topologies and group-valued measures, in: Handbook of measure theory, vol I, II, pp. 703–743 (Ed: E. Pap), North-Holland, Amsterdam, 2002. MR1954626 10.1016/B978-044450263-6/50017-8 H. Weber, FN-Topologies and group-valued measures, in: Handbook of measure theory, vol I, II, pp. 703–743 (Ed: E. Pap), North-Holland, Amsterdam, 2002. MR1954626 10.1016/B978-044450263-6/50017-8
S. Weber, $\perp $-decomposable measures and integrals for Archimedean $t$-conorms $\perp$, J. Math. Anal. Appl. 101 (1984), no. 1, 114–138. MR746230 10.1016/0022-247X(84)90061-1 S. Weber, $\perp $-decomposable measures and integrals for Archimedean $t$-conorms $\perp$, J. Math. Anal. Appl. 101 (1984), no. 1, 114–138. MR746230 10.1016/0022-247X(84)90061-1
