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
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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 |
Vitali-Hahn-Saks theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vitali-Hahn-Saks_theorem&oldid=23107