Vitali-Hahn-Saks theorem

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 .

The Vitali–Hahn–Saks theorem [a7], [a2] says that for any sequence of signed measures which are absolutely continuous with respect to a measure and for which exists for each , the following is true:

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].

R.S. Phillips [a5] and C.E. Rickart [a6] have extended the Vitali–Hahn–Saks theorem to measures with values in a locally convex topological vector space (cf. also Locally convex space).

There are also generalizations to functions defined on orthomodular lattices and with more general properties ([a1], [a4]).

See also Nikodým convergence theorem; Brooks–Jewett theorem.

References

[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