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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b1204901.png" /> be a [[Topological group|topological group]]. A [[Set function|set function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b1204902.png" /> is exhaustive (also called strongly bounded) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b1204903.png" /> for each sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b1204904.png" /> of pairwise disjoint sets from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b1204905.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b1204906.png" /> (cf. also [[Measure|Measure]]). A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b1204907.png" /> of set functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b1204908.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b1204909.png" />, is uniformly exhaustive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049010.png" /> uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049011.png" /> for each sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049012.png" /> of pairwise disjoint sets from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049013.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049014.png" />.
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Being a generalization of the [[Nikodým convergence theorem|Nikodým convergence theorem]], the Brooks–Jewett theorem [[#References|[a1]]] says that for a pointwise-convergent sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049015.png" /> of finitely additive scalar and exhaustive set functions (strongly additive) defined on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049016.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049017.png" />, i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049019.png" />:
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i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049020.png" /> is an additive and exhaustive set function;
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Let $X$ be a [[Topological group|topological group]]. A [[Set function|set function]] $m : \Sigma \rightarrow X$ is exhaustive (also called strongly bounded) if $\operatorname { lim } _ { n \rightarrow \infty } m ( E _ { n } ) = 0$ for each sequence $\{ E _ { n } \}$ of pairwise disjoint sets from the $\sigma$-algebra $\Sigma$ (cf. also [[Measure|Measure]]). A sequence $\{ m_i \}$ of set functions $m _ { i } : \Sigma \rightarrow X$, $i \in \mathbf{N}$, is uniformly exhaustive if $\operatorname { lim } _ { n \rightarrow \infty } m _ { i } ( E _ { n } ) = 0$ uniformly in $i$ for each sequence $\{ E _ { n } \}$ of pairwise disjoint sets from the $\sigma$-algebra $\Sigma$.
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049021.png" /> is uniformly exhaustive.
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Being a generalization of the [[Nikodým convergence theorem|Nikodým convergence theorem]], the Brooks–Jewett theorem [[#References|[a1]]] says that for a pointwise-convergent sequence $\{ m _ { n } \}$ of finitely additive scalar and exhaustive set functions (strongly additive) defined on a $\sigma$-algebra $\Sigma$, i.e. such that $\operatorname { lim } _ { n \rightarrow \infty } m _ { n } ( E ) = m ( E )$, $E \in \Sigma$:
  
There is a generalization of the Brooks–Jewett theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049022.png" />-triangular set functions defined on algebras with some weak <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049023.png" />-conditions (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049024.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049026.png" />-triangular for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049027.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049028.png" /> and
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i) $m$ is an additive and exhaustive set function;
  
<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/b/b120/b120490/b12049029.png" /></td> </tr></table>
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ii) $\{ m _ { n } \}$ is uniformly exhaustive.
  
whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049031.png" />). The following definitions are often used [[#References|[a2]]], [[#References|[a6]]], [[#References|[a5]]]:
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There is a generalization of the Brooks–Jewett theorem for $k$-triangular set functions defined on algebras with some weak $\sigma$-conditions ($m : \Sigma \rightarrow [ 0 , \infty )$ is said to be $k$-triangular for $k \geq 1$ if $m ( \emptyset ) = 0$ and
  
SCP) An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049032.png" /> has the sequential completeness property if each disjoint sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049033.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049034.png" /> has a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049035.png" /> whose union is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049036.png" />.
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\begin{equation*} m ( A ) - k m ( B ) \leq m ( A \bigcup B ) \leq m ( A ) + k m ( B ) \end{equation*}
  
SIP) An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049037.png" /> has the subsequentional interpolation property if for each subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049038.png" /> of each disjoint sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049039.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049040.png" /> there are a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049041.png" /> and a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049042.png" /> such that
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whenever $A , B \in \Sigma$, $A \cap B = \emptyset$). The following definitions are often used [[#References|[a2]]], [[#References|[a6]]], [[#References|[a5]]]:
  
<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/b/b120/b120490/b12049043.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/b/b120/b120490/b12049044.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049045.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
  
According to [[#References|[a5]]]: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049046.png" /> satisfy SIP) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049049.png" />, be a sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049050.png" />-triangular exhaustive set functions. If the limit
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049043.png"/></td> </tr></table>
  
<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/b/b120/b120490/b12049051.png" /></td> </tr></table>
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and $A _ { j } \cap B = \emptyset$ for $j \in \mathbf{N} \backslash \{ j _ { n_k } : k \in \mathbf{N} \}$.
  
exists for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049053.png" /> is exhaustive, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049054.png" /> is uniformly exhaustive and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049055.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049056.png" />-triangular.
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According to [[#References|[a5]]]: Let $\mathcal{A}$ satisfy SIP) and let $\{ m _ { n } \}$, $m _ { n } : {\cal A} \rightarrow [ 0 , + \infty )$, $n \in \mathbf N$, be a sequence of $k$-triangular exhaustive set functions. If the limit
  
There are further generalizations of the Brooks–Jewett theorem, with respect to: the domain of the set functions (orthomodular lattices, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049057.png" />-posets); properties of the set functions; and the range (topological groups, uniform semi-groups, uniform spaces), [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]].
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\begin{equation*} \operatorname { lim } _ { n \rightarrow \infty } m _ { n } ( E ) = m _ { 0 } ( E ) \end{equation*}
 +
 
 +
exists for each $E \in \mathcal{A}$ and $m_0$ is exhaustive, then $\{ m _ { n } \} _ { n = 0 } ^ { \infty }$ is uniformly exhaustive and $m_0$ is $k$-triangular.
 +
 
 +
There are further generalizations of the Brooks–Jewett theorem, with respect to: the domain of the set functions (orthomodular lattices, $D$-posets); properties of the set functions; and the range (topological groups, uniform semi-groups, uniform spaces), [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]].
  
 
It is known that for additive set functions the Brooks–Jewett theorem is equivalent with the [[Nikodým convergence theorem|Nikodým convergence theorem]], and even more with the [[Vitali–Hahn–Saks theorem|Vitali–Hahn–Saks theorem]] [[#References|[a3]]].
 
It is known that for additive set functions the Brooks–Jewett theorem is equivalent with the [[Nikodým convergence theorem|Nikodým convergence theorem]], and even more with the [[Vitali–Hahn–Saks theorem|Vitali–Hahn–Saks theorem]] [[#References|[a3]]].
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Brooks,  R. Jewett,  "On finitely additive vector measures"  ''Proc. Nat. Acad. Sci. USA'' , '''67'''  (1970)  pp. 1294–1298</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">  L. Drewnowski,  "Equivalence of Brooks–Jewett, Vitali–Hahn–Saks and Nikodým theorems"  ''Bull. Acad. Polon. Sci.'' , '''20'''  (1972)  pp. 725–731</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.B. D'Andrea,  P. de Lucia,  "The Brooks–Jewett theorem on an orthomodular lattice"  ''J. Math. Anal. Appl.'' , '''154'''  (1991)  pp. 507–522</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Pap,  "Null-additive set functions" , Kluwer Acad. Publ. &amp;Ister Sci.  (1995)</TD></TR><TR><TD valign="top">[a6]</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><tr><td valign="top">[a1]</td> <td valign="top">  J. Brooks,  R. Jewett,  "On finitely additive vector measures"  ''Proc. Nat. Acad. Sci. USA'' , '''67'''  (1970)  pp. 1294–1298</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">  L. Drewnowski,  "Equivalence of Brooks–Jewett, Vitali–Hahn–Saks and Nikodým theorems"  ''Bull. Acad. Polon. Sci.'' , '''20'''  (1972)  pp. 725–731</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  A.B. D'Andrea,  P. de Lucia,  "The Brooks–Jewett theorem on an orthomodular lattice"  ''J. Math. Anal. Appl.'' , '''154'''  (1991)  pp. 507–522</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  E. Pap,  "Null-additive set functions" , Kluwer Acad. Publ. &amp;Ister Sci.  (1995)</td></tr><tr><td valign="top">[a6]</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>

Revision as of 17:01, 1 July 2020

Let $X$ be a topological group. A set function $m : \Sigma \rightarrow X$ is exhaustive (also called strongly bounded) if $\operatorname { lim } _ { n \rightarrow \infty } m ( E _ { n } ) = 0$ for each sequence $\{ E _ { n } \}$ of pairwise disjoint sets from the $\sigma$-algebra $\Sigma$ (cf. also Measure). A sequence $\{ m_i \}$ of set functions $m _ { i } : \Sigma \rightarrow X$, $i \in \mathbf{N}$, is uniformly exhaustive if $\operatorname { lim } _ { n \rightarrow \infty } m _ { i } ( E _ { n } ) = 0$ uniformly in $i$ for each sequence $\{ E _ { n } \}$ of pairwise disjoint sets from the $\sigma$-algebra $\Sigma$.

Being a generalization of the Nikodým convergence theorem, the Brooks–Jewett theorem [a1] says that for a pointwise-convergent sequence $\{ m _ { n } \}$ of finitely additive scalar and exhaustive set functions (strongly additive) defined on a $\sigma$-algebra $\Sigma$, i.e. such that $\operatorname { lim } _ { n \rightarrow \infty } m _ { n } ( E ) = m ( E )$, $E \in \Sigma$:

i) $m$ is an additive and exhaustive set function;

ii) $\{ m _ { n } \}$ is uniformly exhaustive.

There is a generalization of the Brooks–Jewett theorem for $k$-triangular set functions defined on algebras with some weak $\sigma$-conditions ($m : \Sigma \rightarrow [ 0 , \infty )$ is said to be $k$-triangular for $k \geq 1$ if $m ( \emptyset ) = 0$ and

\begin{equation*} m ( A ) - k m ( B ) \leq m ( A \bigcup B ) \leq m ( A ) + k m ( B ) \end{equation*}

whenever $A , B \in \Sigma$, $A \cap B = \emptyset$). The following definitions are often used [a2], [a6], [a5]:

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

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

According to [a5]: Let $\mathcal{A}$ satisfy SIP) and let $\{ m _ { n } \}$, $m _ { n } : {\cal A} \rightarrow [ 0 , + \infty )$, $n \in \mathbf N$, be a sequence of $k$-triangular exhaustive set functions. If the limit

\begin{equation*} \operatorname { lim } _ { n \rightarrow \infty } m _ { n } ( E ) = m _ { 0 } ( E ) \end{equation*}

exists for each $E \in \mathcal{A}$ and $m_0$ is exhaustive, then $\{ m _ { n } \} _ { n = 0 } ^ { \infty }$ is uniformly exhaustive and $m_0$ is $k$-triangular.

There are further generalizations of the Brooks–Jewett theorem, with respect to: the domain of the set functions (orthomodular lattices, $D$-posets); properties of the set functions; and the range (topological groups, uniform semi-groups, uniform spaces), [a2], [a4], [a5].

It is known that for additive set functions the Brooks–Jewett theorem is equivalent with the Nikodým convergence theorem, and even more with the Vitali–Hahn–Saks theorem [a3].

See also Diagonal theorem.

References

[a1] J. Brooks, R. Jewett, "On finitely additive vector measures" Proc. Nat. Acad. Sci. USA , 67 (1970) pp. 1294–1298
[a2] C. Constantinescu, "Some properties of spaces of measures" Suppl. Atti Sem. Mat. Fis. Univ. Modena , 35 (1991) pp. 1–286
[a3] L. Drewnowski, "Equivalence of Brooks–Jewett, Vitali–Hahn–Saks and Nikodým theorems" Bull. Acad. Polon. Sci. , 20 (1972) pp. 725–731
[a4] A.B. D'Andrea, P. de Lucia, "The Brooks–Jewett theorem on an orthomodular lattice" J. Math. Anal. Appl. , 154 (1991) pp. 507–522
[a5] E. Pap, "Null-additive set functions" , Kluwer Acad. Publ. &Ister Sci. (1995)
[a6] 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:
Brooks-Jewett theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brooks-Jewett_theorem&oldid=22193
This article was adapted from an original article by E. Pap (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article