Difference between revisions of "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].
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 |
Brooks-Jewett theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brooks-Jewett_theorem&oldid=17948