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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" />.
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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:
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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" />;
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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 $\varepsilon > 0$ there exists a $\delta > 0$ such that $| \mu ( E ) | < \varepsilon$ whenever $E \in \Sigma$ and $\lambda ( E ) < \delta$ (cf. also [[Absolute continuity|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 $\lambda ( E ) < \delta$.
  
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
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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:
  
<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>
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i) the limit $\mu$ is also absolutely continuous with respect to this measure, i.e. $\mu \ll \lambda$;
  
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]]].
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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
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\begin{equation*} \operatorname { lim } _ { n \rightarrow \infty } \int _ { E } f _ { n } d \mu = \nu ( E ) \end{equation*}
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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]]).
 
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]]).
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There are also generalizations to functions defined on orthomodular lattices and with more general properties ([[#References|[a1]]], [[#References|[a4]]]).
 
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]].
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See also [[Nikodým convergence theorem]]; [[Brooks–Jewett theorem]].
  
 
====References====
 
====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>
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<table>
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<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>
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</table>

Latest revision as of 07:39, 24 January 2024

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
How to Cite This Entry:
Vitali-Hahn-Saks theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vitali-Hahn-Saks_theorem&oldid=13281
This article was adapted from an original article by E. Pap (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article