View source for Nikodým convergence theorem

<!--This article has been texified automatically. Since there was no Nroff source code for this article, 
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
was used.
If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.

Out of 26 formulas, 24 were replaced by TEX code.-->

{{TEX|semi-auto}}{{TEX|part}}
A theorem [[#References|[a6]]], [[#References|[a7]]], [[#References|[a4]]] saying that for a pointwise convergent sequence $\{ \mu _ { n } \}$ of countably additive measures (cf. [[Measure|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 [[#References|[a2]]], [[#References|[a9]]], [[#References|[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 [[#References|[a3]]], Prop. 8, and A. Grothendieck [[#References|[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|Borel set]]).

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

See also [[Brooks–Jewett theorem|Brooks–Jewett theorem]]; [[Vitali–Hahn–Saks theorem|Vitali–Hahn–Saks theorem]].

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  P. Antosik,   C. Swartz,   "Matrix methods in analysis" , ''Lecture Notes Math.'' , '''1113''' , Springer  (1985)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  C. Constantinescu,   "Some properties of spaces of measures"  ''Suppl. Atti Sem. Mat. Fis. Univ. Modena'' , '''35'''  (1991)  pp. 1–286</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Dieudonné,   "Sur la convergence des suites de mesures de Radon"  ''An. Acad. Brasil. Ci.'' , '''23'''  (1951)  pp. 21–38, 277–282</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators Part I" , Interscience  (1958)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A. Grothendieck,   "Sur les applications linéares faiblement compactes d'espaces du type $C ( K )$"  ''Canad. J. Math.'' , '''5'''  (1953)  pp. 129–173</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  O. Nikodym,   "Sur les suites de functions parfaitement additives d'ensembles abstraits"  ''C.R. Acad. Sci. Paris'' , '''192'''  (1931)  pp. 727</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  O. Nikodym,   "Sur les suites convergentes de functions parfaitement additives d'ensembles abstraits"  ''Monatsh. Math.'' , '''40'''  (1933)  pp. 427–432</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  E. Pap,   "Null-additive set functions" , Kluwer Acad. Publ. &amp;Ister Sci.  (1995)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  W. Schachermayer,   "On some classsical measure-theoretic theorems for non-sigma complete Boolean algebras"  ''Dissert. Math.'' , '''214'''  (1982)  pp. 1–33</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  C. Swartz,   "Introduction to functional analysis" , M. Dekker  (1992)</td></tr>
</table>