View source for Vitali-Hahn-Saks theorem

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v1200301.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v1200302.png" />-algebra (cf. also [[Borel field of sets|Borel field of sets]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v1200303.png" /> be a non-negative [[Set function|set function]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v1200304.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v1200305.png" /> is a normed space. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v1200306.png" /> is absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v1200307.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v1200308.png" />, if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v1200309.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003011.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003013.png" /> (cf. also [[Absolute continuity|Absolute continuity]]). A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003014.png" /> is uniformly absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003015.png" /> if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003016.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003018.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003021.png" />.

The Vitali–Hahn–Saks theorem [[#References|[a7]]], [[#References|[a2]]] says that for any sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003022.png" /> of signed measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003023.png" /> which are absolutely continuous with respect to a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003024.png" /> and for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003025.png" /> exists for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003026.png" />, the following is true:

i) the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003027.png" /> is also absolutely continuous with respect to this measure, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003028.png" />;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003029.png" /> is uniformly absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003030.png" />. This theorem is closely related to integration theory [[#References|[a8]]], [[#References|[a3]]]. Namely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003031.png" /> is a sequence of functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003033.png" /> is the [[Lebesgue measure|Lebesgue measure]], and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003034.png" /></td> </tr></table>

exists for each measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003035.png" />, then the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003036.png" /> is uniformly absolutely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003037.png" />-continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003038.png" /> is absolutely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120030/v12003039.png" />-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>