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A theorem [[#References|[a6]]], [[#References|[a7]]], [[#References|[a4]]] saying that for a pointwise convergent sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n1200801.png" /> of countably additive measures (cf. [[Measure|Measure]]) defined on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n1200802.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n1200803.png" />, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n1200804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n1200805.png" />:
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If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
  
i) the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n1200806.png" /> is a countably additive measure;
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ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n1200807.png" /> is uniformly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n1200809.png" />-additive. As is well-known, the Nikodým convergence theorem for measures fails in general for algebras of sets. But there are convergence theorems in which the initial convergence conditions are imposed on certain subfamilies of a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008010.png" />-algebra; those subfamilies need not be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008011.png" />-algebras. The following definitions are useful [[#References|[a2]]], [[#References|[a9]]], [[#References|[a8]]]:
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A theorem [[#References|[a6]]], [[#References|[a7]]], [[#References|[a4]]] saying that for a pointwise convergent sequence $\{ \mu _ { n } \}$ of countably additive measures (cf. [[Measure|Measure]]) defined on a $\sigma$-algebra $\Sigma$, i.e., $\operatorname { lim } _ { n \rightarrow \infty } \mu _ { n } ( E ) = \mu ( E )$, $E \in \Sigma$:
  
SCP) An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008012.png" /> has the sequential completeness property if each disjoint sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008013.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008014.png" /> has a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008015.png" /> whose union is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008016.png" />.
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i) the limit $m$ is a countably additive measure;
  
SIP) An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008017.png" /> has the subsequentional interpolation property if for each subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008018.png" /> of each disjoint sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008019.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008020.png" /> there are a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008021.png" /> and a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008022.png" /> such that
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ii) $\{ \mu _ { n } \}$ is uniformly $\sigma$-additive. As is well-known, the Nikodým convergence theorem for measures fails in general for algebras of sets. But there are convergence theorems in which the initial convergence conditions are imposed on certain subfamilies of a given $\sigma$-algebra; those subfamilies need not be $\sigma$-algebras. The following definitions are useful [[#References|[a2]]], [[#References|[a9]]], [[#References|[a8]]]:
  
<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/n/n120/n120080/n12008023.png" /></td> </tr></table>
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SCP) An algebra $\mathcal{A}$ has the sequential completeness property if each disjoint sequence $\{ E _ { n } \}$ from $\mathcal{A}$ has a subsequence $\{ E _ { n_j} \}$ whose union is in $\mathcal{A}$.
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008024.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008025.png" />.
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SIP) An algebra $\mathcal{A}$ has the subsequentional interpolation property if for each subsequence $\{ A _ { j n } \}$ of each disjoint sequence $\{ A _ { j } \}$ from $\mathcal{A}$ there are a subsequence $\{ A _ { j n _ { k } } \}$ and a set $B \in \mathcal{A}$ such that
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$$A _ { j n _ { k } } \subset B, \quad k \in \mathbf{N}$$
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and $A _ { j } \cap B = \emptyset$ for $j \in \mathbf{N} \backslash \{ j _ { n_k } : k \in \mathbf{N} \}$.
  
 
The Nikodým convergence theorem holds on algebras with SCP) and SIP).
 
The Nikodým convergence theorem holds on algebras with SCP) and SIP).
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====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">  C. Constantinescu,  "Some properties of spaces of measures"  ''Suppl. Atti Sem. Mat. Fis. Univ. Modena'' , '''35'''  (1991)  pp. 1–286</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Dieudonné,  "Sur la convergence des suites de mesures de Radon"  ''An. Acad. Brasil. Ci.'' , '''23'''  (1951)  pp. 21–38, 277–282</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators Part I" , Interscience  (1958)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Grothendieck,  "Sur les applications linéares faiblement compactes d'espaces du type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120080/n12008026.png" />"  ''Canad. J. Math.'' , '''5'''  (1953)  pp. 129–173</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  O. Nikodym,  "Sur les suites de functions parfaitement additives d'ensembles abstraits"  ''C.R. Acad. Sci. Paris'' , '''192'''  (1931)  pp. 727</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  O. Nikodym,  "Sur les suites convergentes de functions parfaitement additives d'ensembles abstraits"  ''Monatsh. Math.'' , '''40'''  (1933)  pp. 427–432</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  E. Pap,  "Null-additive set functions" , Kluwer Acad. Publ. &amp;Ister Sci.  (1995)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  W. Schachermayer,  "On some classsical measure-theoretic theorems for non-sigma complete Boolean algebras"  ''Dissert. Math.'' , '''214'''  (1982)  pp. 1–33</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  C. Swartz,  "Introduction to functional analysis" , M. Dekker  (1992)</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">  C. Constantinescu,  "Some properties of spaces of measures"  ''Suppl. Atti Sem. Mat. Fis. Univ. Modena'' , '''35'''  (1991)  pp. 1–286</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Dieudonné,  "Sur la convergence des suites de mesures de Radon"  ''An. Acad. Brasil. Ci.'' , '''23'''  (1951)  pp. 21–38, 277–282</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators Part I" , Interscience  (1958)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A. Grothendieck,  "Sur les applications linéares faiblement compactes d'espaces du type $C ( K )$"  ''Canad. J. Math.'' , '''5'''  (1953)  pp. 129–173</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  O. Nikodym,  "Sur les suites de functions parfaitement additives d'ensembles abstraits"  ''C.R. Acad. Sci. Paris'' , '''192'''  (1931)  pp. 727</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  O. Nikodym,  "Sur les suites convergentes de functions parfaitement additives d'ensembles abstraits"  ''Monatsh. Math.'' , '''40'''  (1933)  pp. 427–432</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  E. Pap,  "Null-additive set functions" , Kluwer Acad. Publ. &amp;Ister Sci.  (1995)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  W. Schachermayer,  "On some classsical measure-theoretic theorems for non-sigma complete Boolean algebras"  ''Dissert. Math.'' , '''214'''  (1982)  pp. 1–33</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  C. Swartz,  "Introduction to functional analysis" , M. Dekker  (1992)</td></tr>
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</table>

Latest revision as of 07:42, 24 November 2023

A theorem [a6], [a7], [a4] saying that for a pointwise convergent sequence $\{ \mu _ { n } \}$ of countably additive measures (cf. Measure) defined on a $\sigma$-algebra $\Sigma$, i.e., $\operatorname { lim } _ { n \rightarrow \infty } \mu _ { n } ( E ) = \mu ( E )$, $E \in \Sigma$:

i) the limit $m$ is a countably additive measure;

ii) $\{ \mu _ { n } \}$ is uniformly $\sigma$-additive. As is well-known, the Nikodým convergence theorem for measures fails in general for algebras of sets. But there are convergence theorems in which the initial convergence conditions are imposed on certain subfamilies of a given $\sigma$-algebra; those subfamilies need not be $\sigma$-algebras. The following definitions are useful [a2], [a9], [a8]:

SCP) An algebra $\mathcal{A}$ has the sequential completeness property if each disjoint sequence $\{ E _ { n } \}$ from $\mathcal{A}$ has a subsequence $\{ E _ { n_j} \}$ whose union is in $\mathcal{A}$.

SIP) An algebra $\mathcal{A}$ has the subsequentional interpolation property if for each subsequence $\{ A _ { j n } \}$ of each disjoint sequence $\{ A _ { j } \}$ from $\mathcal{A}$ there are a subsequence $\{ A _ { j n _ { k } } \}$ and a set $B \in \mathcal{A}$ such that

$$A _ { j n _ { k } } \subset B, \quad k \in \mathbf{N}$$

and $A _ { j } \cap B = \emptyset$ for $j \in \mathbf{N} \backslash \{ j _ { n_k } : k \in \mathbf{N} \}$.

The Nikodým convergence theorem holds on algebras with SCP) and SIP).

A famous result of J. Dieudonné [a3], Prop. 8, and A. Grothendieck [a5], p. 150, states that for compact metric spaces, respectively locally compact spaces, convergence of a sequence of regular Borel measures on every open set implies convergence on all Borel sets (cf. also Borel set).

Many related results can be found in [a1], [a8], where the method of diagonal theorems is used instead of the commonly used Baire category theorem (see [a4], [a10] and Diagonal theorem).

See also Brooks–Jewett theorem; Vitali–Hahn–Saks theorem.

References

[a1] P. Antosik, C. Swartz, "Matrix methods in analysis" , Lecture Notes Math. , 1113 , Springer (1985)
[a2] C. Constantinescu, "Some properties of spaces of measures" Suppl. Atti Sem. Mat. Fis. Univ. Modena , 35 (1991) pp. 1–286
[a3] J. Dieudonné, "Sur la convergence des suites de mesures de Radon" An. Acad. Brasil. Ci. , 23 (1951) pp. 21–38, 277–282
[a4] N. Dunford, J.T. Schwartz, "Linear operators Part I" , Interscience (1958)
[a5] A. Grothendieck, "Sur les applications linéares faiblement compactes d'espaces du type $C ( K )$" Canad. J. Math. , 5 (1953) pp. 129–173
[a6] O. Nikodym, "Sur les suites de functions parfaitement additives d'ensembles abstraits" C.R. Acad. Sci. Paris , 192 (1931) pp. 727
[a7] O. Nikodym, "Sur les suites convergentes de functions parfaitement additives d'ensembles abstraits" Monatsh. Math. , 40 (1933) pp. 427–432
[a8] E. Pap, "Null-additive set functions" , Kluwer Acad. Publ. &Ister Sci. (1995)
[a9] W. Schachermayer, "On some classsical measure-theoretic theorems for non-sigma complete Boolean algebras" Dissert. Math. , 214 (1982) pp. 1–33
[a10] C. Swartz, "Introduction to functional analysis" , M. Dekker (1992)
How to Cite This Entry:
Nikodým convergence theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nikod%C3%BDm_convergence_theorem&oldid=15798
This article was adapted from an original article by E. Pap (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article