Nikodým convergence theorem

A theorem [a6], [a7], [a4] saying that for a pointwise convergent sequence $\{ \mu _ { n } \}$ of countably additive measures (cf. Measure) defined on a $\sigma$-algebra $\Sigma$, i.e., $\operatorname { lim } _ { n \rightarrow \infty } \mu _ { n } ( E ) = \mu ( E )$, $E \in \Sigma$:

i) the limit $m$ is a countably additive measure;

ii) $\{ \mu _ { n } \}$ is uniformly $\sigma$-additive. As is well-known, the Nikodým convergence theorem for measures fails in general for algebras of sets. But there are convergence theorems in which the initial convergence conditions are imposed on certain subfamilies of a given $\sigma$-algebra; those subfamilies need not be $\sigma$-algebras. The following definitions are useful [a2], [a9], [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 subsequentional 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

$$A _ { j n _ { k } } \subset B, \quad k \in \mathbf{N}$$

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

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

A famous result of J. Dieudonné [a3], Prop. 8, and A. Grothendieck [a5], p. 150, states that for compact metric spaces, respectively locally compact spaces, convergence of a sequence of regular Borel measures on every open set implies convergence on all Borel sets (cf. also Borel set).

Many related results can be found in [a1], [a8], where the method of diagonal theorems is used instead of the commonly used Baire category theorem (see [a4], [a10] and Diagonal theorem).

See also Brooks–Jewett theorem; Vitali–Hahn–Saks theorem.

References

[a1]	P. Antosik, C. Swartz, "Matrix methods in analysis" , Lecture Notes Math. , 1113 , Springer (1985)
[a2]	C. Constantinescu, "Some properties of spaces of measures" Suppl. Atti Sem. Mat. Fis. Univ. Modena , 35 (1991) pp. 1–286
[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]	A. Grothendieck, "Sur les applications linéares faiblement compactes d'espaces du type $C ( K )$" Canad. J. Math. , 5 (1953) pp. 129–173
[a6]	O. Nikodym, "Sur les suites de functions parfaitement additives d'ensembles abstraits" C.R. Acad. Sci. Paris , 192 (1931) pp. 727
[a7]	O. Nikodym, "Sur les suites convergentes de functions parfaitement additives d'ensembles abstraits" Monatsh. Math. , 40 (1933) pp. 427–432
[a8]	E. Pap, "Null-additive set functions" , Kluwer Acad. Publ. &Ister Sci. (1995)
[a9]	W. Schachermayer, "On some classsical measure-theoretic theorems for non-sigma complete Boolean algebras" Dissert. Math. , 214 (1982) pp. 1–33
[a10]	C. Swartz, "Introduction to functional analysis" , M. Dekker (1992)