# Brooks-Jewett theorem

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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].