Nikodým boundedness theorem

A theorem [a5], [a4], saying that a family $\mathcal{M}$ of countably additive signed measures $m$ (cf. Measure) defined on a $\sigma$-algebra $\Sigma$ and pointwise bounded, i.e. for each $E \in \Sigma$ there exists a number $M _ { E } > 0$ such that

\begin{equation*} | m ( E ) | < M _ { E } , \quad m \in \mathcal{M}, \end{equation*}

is uniformly bounded, i.e. there exists a number $M > 0$ such that

\begin{equation*} | m ( E ) | < M , \quad m \in \mathcal{M} , E \in \Sigma. \end{equation*}

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 $\sigma$-algebra; those subfamilies need not be $\sigma$-algebras. The following definitions are useful [a2], [a7], [a8]:

SCP) An algebra $\mathcal{A}$ has the sequential completeness property if each disjoint sequence $\{ E _ { n } \}$ from $\mathcal{A}$ has a subsequence $\{ E _ { n_j} \}$ whose union is in $\mathcal{A}$

SIP) An algebra $\mathcal{A}$ has the subsequential interpolation property if for each subsequence $\{ A _ { j n } \}$ of each disjoint sequence $\{ A _ { j } \}$ from $\mathcal{A}$ there are a subsequence $\{ A _ { j n _ { k } } \}$ and a set $B \in \mathcal{A}$ such that

\begin{equation*} A _ { j_{n _ { k } }} \subset B , \quad k \in \bf N \end{equation*}

and $A _ { j } \cap B = \emptyset$ for $j \in \mathbf{N} \backslash \{ j _ { n_k } : k \in \mathbf{N} \}$.

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