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A theorem [[#References|[a5]]], [[#References|[a4]]], saying that a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n1200701.png" /> of countably additive signed measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n1200702.png" /> (cf. [[Measure|Measure]]) defined on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n1200703.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n1200704.png" /> and pointwise bounded, i.e. for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n1200705.png" /> there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n1200706.png" /> such that
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<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/n120070/n1200707.png" /></td> </tr></table>
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is uniformly bounded, i.e. there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n1200708.png" /> such that
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A theorem [[#References|[a5]]], [[#References|[a4]]], saying that a family $\mathcal{M}$ of countably additive signed measures $m$ (cf. [[Measure|Measure]]) defined on a $\sigma$-algebra $\Sigma$ and pointwise bounded, i.e. for each $E \in \Sigma$ there exists a number $M _ { E } > 0$ such that
  
<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/n120070/n1200709.png" /></td> </tr></table>
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\begin{equation*} | m ( E ) | < M _ { E } , \quad m \in \mathcal{M}, \end{equation*}
  
As is well-known, the Nikodým boundedness theorem for measures fails in general for algebras of sets. But there are uniform boundedness theorems in which the initial boundedness conditions are imposed on certain subfamilies of a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n12007010.png" />-algebra; those subfamilies need not be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n12007011.png" />-algebras. The following definitions are useful [[#References|[a2]]], [[#References|[a7]]], [[#References|[a8]]]:
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is uniformly bounded, i.e. there exists a number $M > 0$ such that
  
SCP) An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n12007012.png" /> has the sequential completeness property if each disjoint sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n12007013.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n12007014.png" /> has a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n12007015.png" /> whose union is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n12007016.png" />
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\begin{equation*} | m ( E ) | < M , \quad m \in \mathcal{M} , E \in \Sigma. \end{equation*}
  
SIP) An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n12007017.png" /> has the subsequential interpolation property if for each subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n12007018.png" /> of each disjoint sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n12007019.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n12007020.png" /> there are a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n12007021.png" /> and a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n12007022.png" /> such that
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As is well-known, the Nikodým boundedness theorem for measures fails in general for algebras of sets. But there are uniform boundedness theorems in which the initial boundedness 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|[a7]]], [[#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/n120070/n12007023.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/n120070/n12007024.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120070/n12007025.png" />.
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SIP) An algebra $\mathcal{A}$ has the subsequential 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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\begin{equation*} A _ { j_{n _ { k } }} \subset B , \quad k \in \bf N \end{equation*}
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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 boundedness theorem holds on algebras with SCP) and SIP).
 
The Nikodým boundedness 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,  "On Nikodym's boundedness theorem"  ''Libertas Math.'' , '''1'''  (1981)  pp. 51–73</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">  O. Nikodym,  "Sur les familles bornées de functions parfaitement additives d'ensembles abstraits"  ''Monatsh. Math.'' , '''40'''  (1933)  pp. 418–426</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E. Pap,  "Null-additive set functions" , Kluwer Acad. Publ. &amp;Ister Sci.  (1995)</TD></TR><TR><TD valign="top">[a7]</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">[a8]</TD> <TD valign="top">  H. Weber,  "Compactness in spaces of group-valued contents, the Vitali–Hahn–Saks theorem and the Nikodym's boundedness theorem"  ''Rocky Mtn. J. Math.'' , '''16'''  (1986)  pp. 253–275</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,  "On Nikodym's boundedness theorem"  ''Libertas Math.'' , '''1'''  (1981)  pp. 51–73</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">  O. Nikodym,  "Sur les familles bornées de functions parfaitement additives d'ensembles abstraits"  ''Monatsh. Math.'' , '''40'''  (1933)  pp. 418–426</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  E. Pap,  "Null-additive set functions" , Kluwer Acad. Publ. &amp;Ister Sci.  (1995)</td></tr><tr><td valign="top">[a7]</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">[a8]</td> <td valign="top">  H. Weber,  "Compactness in spaces of group-valued contents, the Vitali–Hahn–Saks theorem and the Nikodym's boundedness theorem"  ''Rocky Mtn. J. Math.'' , '''16'''  (1986)  pp. 253–275</td></tr>
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</table>

Latest revision as of 20:19, 25 January 2024

A theorem [a5], [a4], saying that a family $\mathcal{M}$ of countably additive signed measures $m$ (cf. Measure) defined on a $\sigma$-algebra $\Sigma$ and pointwise bounded, i.e. for each $E \in \Sigma$ there exists a number $M _ { E } > 0$ such that

\begin{equation*} | m ( E ) | < M _ { E } , \quad m \in \mathcal{M}, \end{equation*}

is uniformly bounded, i.e. there exists a number $M > 0$ such that

\begin{equation*} | m ( E ) | < M , \quad m \in \mathcal{M} , E \in \Sigma. \end{equation*}

As is well-known, the Nikodým boundedness theorem for measures fails in general for algebras of sets. But there are uniform boundedness theorems in which the initial boundedness conditions are imposed on certain subfamilies of a given $\sigma$-algebra; those subfamilies need not be $\sigma$-algebras. The following definitions are useful [a2], [a7], [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 subsequential 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

\begin{equation*} A _ { j_{n _ { k } }} \subset B , \quad k \in \bf N \end{equation*}

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

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

A famous theorem of J. Dieudonné [a3] states that for compact metric spaces the pointwise boundedness of a family of regular Borel measures on open sets implies its uniform boundedness on all Borel sets. There are further generalizations of this theorem [a6].

See also Nikodým convergence theorem; Diagonal theorem.

References

[a1] P. Antosik, C. Swartz, "Matrix methods in analysis" , Lecture Notes Math. , 1113 , Springer (1985)
[a2] C. Constantinescu, "On Nikodym's boundedness theorem" Libertas Math. , 1 (1981) pp. 51–73
[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] O. Nikodym, "Sur les familles bornées de functions parfaitement additives d'ensembles abstraits" Monatsh. Math. , 40 (1933) pp. 418–426
[a6] E. Pap, "Null-additive set functions" , Kluwer Acad. Publ. &Ister Sci. (1995)
[a7] W. Schachermayer, "On some classsical measure-theoretic theorems for non-sigma complete Boolean algebras" Dissert. Math. , 214 (1982) pp. 1–33
[a8] H. Weber, "Compactness in spaces of group-valued contents, the Vitali–Hahn–Saks theorem and the Nikodym's boundedness theorem" Rocky Mtn. J. Math. , 16 (1986) pp. 253–275
How to Cite This Entry:
Nikodým boundedness theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nikod%C3%BDm_boundedness_theorem&oldid=22847
This article was adapted from an original article by E. Pap (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article