# Vitali-Hahn-Saks theorem

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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]).

How to Cite This Entry:
Vitali-Hahn-Saks theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vitali-Hahn-Saks_theorem&oldid=50389
This article was adapted from an original article by E. Pap (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article