Duality and integral representation for excessive measures

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meant in the above sense We consider и to be fixed for the rest of this section, therefore we suppress it m further notation and, e g, write ^ instead of ^„ Finally let us defme Exc^ ={тбЕхс Цт, w)< oo} It is immediate from the nition that (^, u) IS an order-unit space (with respect to the natural order) as defined in Sect 1, hence a normed vector space wrt the order unit norm ||/i||„ =mf{)'>0 —yu^h^yu} for he^ The dual ^' of ^ becomes a base-norm space wrt the induced dual norm „|| || Note that the positive cone p^ m Qj' consists of all (II • L-contmuous) linear functional which are positive on pB

Recall the definition of the energy functional from Sect 1, for meExc^ we extend L{m,') to В by hnearity Then L(m,') is a positive hnear functional on Q) which is sequentially continuous from below o?i e9^, le if (s„) increases to s m =5^ then L(m, s„) increases to L{m, s)

The subsequent result extends a theorem by Boboc et al for resolvents m strong duality, (cf [7, Proposition 12 3]), as well a remark by Mokobodzki for Ray resolvents (cf [29]) Besides, it can be derived as a special case of Theorem 5 8m [8] on lattice cones

( 2 . 2 ) Theorem. There is a l — l correspondence between Exc„ and the set of func- tionals in pB which are sequentially continuous from below on ^^ The dence IS given by m\-^L{m •) for тбЕхс„

Proof Let /lep^ be sequentially continuous from below Then since iu{u)< сю, m{f) =iu{Vf) defines a positive linear functional on the vector space generated by {fepS' Vf^nu for some wgN}, m is, moreover, sequentially continuous from below, hence a measure on <f, m is a-fimte since for a ^>0 sth Vq^u (see 2 1) one obtains m{q) = iii(Vq)^ fi{u)< со, m is moreover excessive because for fepi s th Vf^u,m satisfies ocmVJif) = m{(xVJ) = fi{aV^ Vf) which increases to fi{Vf) = m{f) since in is sequentially continuous from below on 5^ more, let (F/„) increase to se6^^, then L(m,s) = îlimw(/„) = îlim/i(K/J = iU(s),

n n

m particular, one obtains L{m, w) = /i(w)< oo, thus тбЕхс„

On the other hand, given meExc^, then for seB, /i(s) =L{m,s) defines a positive (hence ace to 1 1 bounded) hnear functional on ^, which is sequentially continuous from below on 6^^, according to the remark above the statement of (2 2) П

Consider on В the balayage order " < " induced by the cone 5^

( 23 ) /i-<v if jm(5)^v(s) for all s g ^„

It obviously extends what is known as "balayage order" on measures to the space В It is immediate that for p, rjeExc^^ we have p^rj lî and only if L{p,')< L{ri,') Consequently we have

( 2 . 4 ) Corollary. The sets pB'^ ={pepQ) p sequentially continuous from below on ^} and Exc„ are order isomorphic wrt the balayage order on pB and the natural order on Exc„

Reformulating (2 2) gives the first part of the relationship between 6^ and Exc which was sketched in the introduction

( 2 . 5 ) Corollary. For m g Exc, L{m,') defines a positive linear Ш+-valued functional on ^ which is sequentially continuous from below, and which satisfies L[m, u) < go