View source for Brooks-Jewett theorem

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" />.

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" />:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049020.png" /> is an additive and exhaustive set function;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120490/b12049021.png" /> is uniformly exhaustive.

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

<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>

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]]]:

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" />.

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

<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>

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" />.

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

<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>

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.

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]]].

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]]].

See also [[Diagonal theorem|Diagonal theorem]].

====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>