Nikodým boundedness theorem

A theorem [a5], [a4], saying that a family of countably additive signed measures (cf. Measure) defined on a -algebra and pointwise bounded, i.e. for each there exists a number such that

is uniformly bounded, i.e. there exists a number such that

As is well-known, the Nikodým boundedness theorem for measures fails in general for algebras of sets. But there are uniform boundedness theorems in which the initial boundedness conditions are imposed on certain subfamilies of a given -algebra; those subfamilies need not be -algebras. The following definitions are useful [a2], [a7], [a8]:

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 subsequential interpolation property if for each subsequence of each disjoint sequence from there are a subsequence and a set such that

and for .

The Nikodým boundedness theorem holds on algebras with SCP) and SIP).

A famous theorem of J. Dieudonné [a3] states that for compact metric spaces the pointwise boundedness of a family of regular Borel measures on open sets implies its uniform boundedness on all Borel sets. There are further generalizations of this theorem [a6].

See also Nikodým convergence theorem; Diagonal theorem.

References

[a1]	P. Antosik, C. Swartz, "Matrix methods in analysis" , Lecture Notes Math. , 1113 , Springer (1985)
[a2]	C. Constantinescu, "On Nikodym's boundedness theorem" Libertas Math. , 1 (1981) pp. 51–73
[a3]	J. Dieudonné, "Sur la convergence des suites de mesures de Radon" An. Acad. Brasil. Ci. , 23 (1951) pp. 21–38, 277–282
[a4]	N. Dunford, J.T. Schwartz, "Linear operators Part I" , Interscience (1958)
[a5]	O. Nikodym, "Sur les familles bornées de functions parfaitement additives d'ensembles abstraits" Monatsh. Math. , 40 (1933) pp. 418–426
[a6]	E. Pap, "Null-additive set functions" , Kluwer Acad. Publ. &Ister Sci. (1995)
[a7]	W. Schachermayer, "On some classsical measure-theoretic theorems for non-sigma complete Boolean algebras" Dissert. Math. , 214 (1982) pp. 1–33
[a8]	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