On extension of submeasures
Ivan Dobrakov · Czech digital mathematics library · 1984
DOBRAKOV Let 91 be a ring of subsets of a non-empty set T. According to Definition 1 in [1] we say that a set function /i: 91->[0, +oo) is a submeasure if it is 1) monotone, 2) continuous: A n e 91, n = 1, 2, ..., and A n \0 implies ju(A n )->0, and subadditively continuous: For every Ae9l and e > 0 there is a 6 > 0 such that B e 91 and JU(B)0, then JU(A")->JU(A).Similarly, the uniform subadditive continuity is equivalent to the following one: 3u)*: for each e > 0 there is a 6 > 0 such that A, B e 91 and /z(AAB) |JU(A) -/i(B)| [0, +oo] is exhaustive if ju(A n )->0 for each infinite sequence A n e 91, n = 1, 2, ... of pairwise disjoint sets.In Theorem 18 in [1] we proved, see also [3] for another proof, that a uniform, subadditive or additive submeasure JU: 91-»[0, + oo) has a unique extension of the same type to o(9l) -the a-ring generated by 91, if and only if it is exhaustive.Two additional, rather clumsy, conditions were needed to obtain the extension theorem for non-uniform submeasures.In this note, using a more transparent approach we show that these conditions may be replaced by the following: (ii) below, and A n e9l, n = l,2, ... and /i(A n AA m )->0 as n, m->oo implies that /i(A n ) -/i(A m )->0 as n, m-»oo.We start with a set function JU: 91->[0, +oo) having the following properties: (i) JU is monotone and JU(0) = O, (ii) JU has the pseudometric generating property, briefly the (p.g.p.), see Theorem 1 in [2]: For each e>0 there is a 0 such that A, B e9i, \i(A), \i(B)<b implies /i(AuB)<£, and (iii) \x has the Fatou property, briefly the (F.p.): A, A n e 91, n = l,2, ... and A n /A implies ju(A n )/V(A).