Let be a -algebra (cf. also Borel field of sets). Let be a non-negative set function and let , where is a normed space. One says that is absolutely continuous with respect to , denoted by , if for every there exists a such that whenever and (cf. also Absolute continuity). A sequence is uniformly absolutely continuous with respect to if for every there exists a such that whenever , and .
i) the limit is also absolutely continuous with respect to this measure, i.e. ;
ii) is uniformly absolutely continuous with respect to . This theorem is closely related to integration theory [a8], [a3]. Namely, if is a sequence of functions from , where is the Lebesgue measure, and
exists for each measurable set , then the sequence is uniformly absolutely -continuous and is absolutely -continuous, [a3].
|[a1]||P. Antosik, C. Swartz, "Matrix methods in analysis" , Lecture Notes Math. , 1113 , Springer (1985)|
|[a2]||N. Dunford, J.T. Schwartz, "Linear operators, Part I" , Interscience (1958)|
|[a3]||H. Hahn, "Über Folgen linearer Operationen" Monatsh. Math. Physik , 32 (1922) pp. 3–88|
|[a4]||E. Pap, "Null-additive set functions" , Kluwer Acad. Publ. &Ister Sci. (1995)|
|[a5]||R.S. Phillips, "Integration in a convex linear topological space" Trans. Amer. Math. Soc. , 47 (1940) pp. 114–145|
|[a6]||C.E. Rickart, "Integration in a convex linear topological space" Trans. Amer. Math. Soc. , 52 (1942) pp. 498–521|
|[a7]||S. Saks, "Addition to the note on some functionals" Trans. Amer. Math. Soc. , 35 (1933) pp. 967–974|
|[a8]||G. Vitali, "Sull' integrazione per serie" Rend. Circ. Mat. Palermo , 23 (1907) pp. 137–155|
Vitali-Hahn-Saks theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vitali-Hahn-Saks_theorem&oldid=13281