A subset
of an algebra
of subsets of a set
has property
if each
-pointwise bounded sequence of the Banach space
is bounded in
, where
is the Banach space of real or complex bounded finitely additive measures defined on
endowed with the variation norm.
has property
[
] if for each bounded sequence [if for each sequence] in
the
-pointwise convergence implies its weak convergence.
has property
[
or
] if every increasing covering
of
contains a set
with property
[
or
], and
has property
[
or
] if every increasing web
of
contains a strand
formed by elements
with property
[
or
] for every
. The classical theorems of Nikodým–Grothendieck, Valdivia, Grothendieck and Vitali–Hahn–Saks say, respectively, that every
-algebra has properties
,
,
and
. Valdivia’s theorem was obtained through theorems of barrelled spaces. Recently, it has been proved that every
-algebra has property
and several applications of this strong Nikodým type property have been provided. In this survey paper we obtain a proof of the property
of a
-algebra independent of the theory of locally convex barrelled spaces which depends on elementary basic results of Measure theory and Banach space theory. Moreover we prove that a subset
of an algebra
has property
if and only if
has property
and
has property
.
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