Brooks-Jewett theorem

Let be a topological group. A set function is exhaustive (also called strongly bounded) if for each sequence of pairwise disjoint sets from the -algebra (cf. also Measure). A sequence of set functions , , is uniformly exhaustive if uniformly in for each sequence of pairwise disjoint sets from the -algebra .

Being a generalization of the Nikodým convergence theorem, the Brooks–Jewett theorem [a1] says that for a pointwise-convergent sequence of finitely additive scalar and exhaustive set functions (strongly additive) defined on a -algebra , i.e. such that , :

i) is an additive and exhaustive set function;

ii) is uniformly exhaustive.

There is a generalization of the Brooks–Jewett theorem for -triangular set functions defined on algebras with some weak -conditions ( is said to be -triangular for if and

whenever , ). The following definitions are often used [a2], [a6], [a5]:

SCP) An algebra has the sequential completeness property if each disjoint sequence from has a subsequence whose union is in .

SIP) An algebra has the subsequentional interpolation property if for each subsequence of each disjoint sequence from there are a subsequence and a set such that

and for .

According to [a5]: Let satisfy SIP) and let , , , be a sequence of -triangular exhaustive set functions. If the limit

exists for each and is exhaustive, then is uniformly exhaustive and is -triangular.

There are further generalizations of the Brooks–Jewett theorem, with respect to: the domain of the set functions (orthomodular lattices, -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