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Let $\Sigma$ be a $\sigma$-algebra (cf. also [[Borel field of sets|Borel field of sets]]). Let $\lambda : \Sigma \rightarrow [ 0 , + \infty ]$ be a non-negative [[Set function|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 $\varepsilon &gt; 0$ there exists a $\delta &gt; 0$ such that $| \mu ( E ) | &lt; \varepsilon$ whenever $E \in \Sigma$ and $\lambda ( E ) &lt; \delta$ (cf. also [[Absolute continuity|Absolute continuity]]). A sequence $\{ \mu _ { n } \}$ is uniformly absolutely continuous with respect to $\lambda$ if for every $\varepsilon &gt; 0$ there exists a $\delta &gt; 0$ such that $| \mu _ { n } ( E ) | &lt; \varepsilon$ whenever $E \in \Sigma$, $n \in \mathbf N$ and $\lambda ( E ) &lt; \delta$.

The Vitali–Hahn–Saks theorem [[#References|[a7]]], [[#References|[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 [[#References|[a8]]], [[#References|[a3]]]. Namely, if $\{ f _ { n } \}$ is a sequence of functions from $L _ { 1 } ( [ 0,1 ] )$, where $\mu$ is the [[Lebesgue measure|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, [[#References|[a3]]].

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

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

See also [[Nikodým convergence theorem|Nikodým convergence theorem]]; [[Brooks–Jewett theorem|Brooks–Jewett theorem]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  P. Antosik,   C. Swartz,   "Matrix methods in analysis" , ''Lecture Notes Math.'' , '''1113''' , Springer  (1985)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators, Part I" , Interscience  (1958)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  H. Hahn,   "Über Folgen linearer Operationen"  ''Monatsh. Math. Physik'' , '''32'''  (1922)  pp. 3–88</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  E. Pap,   "Null-additive set functions" , Kluwer Acad. Publ. &amp;Ister Sci.  (1995)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  R.S. Phillips,   "Integration in a convex linear topological space"  ''Trans. Amer. Math. Soc.'' , '''47'''  (1940)  pp. 114–145</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  C.E. Rickart,   "Integration in a convex linear topological space"  ''Trans. Amer. Math. Soc.'' , '''52'''  (1942)  pp. 498–521</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  S. Saks,   "Addition to the note on some functionals"  ''Trans. Amer. Math. Soc.'' , '''35'''  (1933)  pp. 967–974</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  G. Vitali,   "Sull' integrazione per serie"  ''Rend. Circ. Mat. Palermo'' , '''23'''  (1907)  pp. 137–155</td></tr></table>