Vitali-Hahn-Saks theorem

Let $\Sigma$ be a $\sigma$-algebra (cf. also Borel field of sets). Let $\lambda : \Sigma \rightarrow [ 0 , + \infty ]$ be a non-negative set function and let $\mu : \Sigma \rightarrow X$, where $X$ is a normed space. One says that $\mu$ is absolutely continuous with respect to $\lambda$, denoted by $\mu \ll \lambda$, if for every $\varepsilon > 0$ there exists a $\delta > 0$ such that $| \mu ( E ) | < \varepsilon$ whenever $E \in \Sigma$ and $\lambda ( E ) < \delta$ (cf. also Absolute continuity). A sequence $\{ \mu _ { n } \}$ is uniformly absolutely continuous with respect to $\lambda$ if for every $\varepsilon > 0$ there exists a $\delta > 0$ such that $| \mu _ { n } ( E ) | < \varepsilon$ whenever $E \in \Sigma$, $n \in \mathbf N$ and $\lambda ( E ) < \delta$.

The Vitali–Hahn–Saks theorem [a7], [a2] says that for any sequence $\{ \mu _ { n } \}$ of signed measures $\mu _ { n }$ which are absolutely continuous with respect to a measure $\lambda$ and for which $\operatorname { lim } _ { n \rightarrow \infty } \mu _ { n } ( E ) = \mu ( E )$ exists for each $E \in \Sigma$, the following is true:

i) the limit $\mu$ is also absolutely continuous with respect to this measure, i.e. $\mu \ll \lambda$;

ii) $\{ \mu _ { n } \}$ is uniformly absolutely continuous with respect to $\lambda$. This theorem is closely related to integration theory [a8], [a3]. Namely, if $\{ f _ { n } \}$ is a sequence of functions from $L _ { 1 } ( [ 0,1 ] )$, where $\mu$ is the Lebesgue measure, and

\begin{equation*} \operatorname { lim } _ { n \rightarrow \infty } \int _ { E } f _ { n } d \mu = \nu ( E ) \end{equation*}

exists for each measurable set $E$, then the sequence $\{ \int f _ { n } d \mu \}$ is uniformly absolutely $\mu$-continuous and $\nu$ is absolutely $\mu$-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