X e V+ such that X is an extreme point of the convex set of charges in V which coincide with A on a subalgebra of 3t. Then each charge p e V+ has a unique.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL Volume 96. Number
1. Januar,
SOCIETY
1986
DECOMPOSITION OF CHARGES AND MEASURES HARALD LUSCHGY Abstract. By means of a characterization of the band generated by a set of charges on an algebra of sets one can obtain unified proofs of some decomposition theorems for charges and measures.
1. Introduction. Let 31 be an algebra of subsets of a set S2.By ba(3t) we denote the order complete vector lattice of all bounded, additive, real functions on 3Í and elements of its positive cone ba( 21) + are referred to as charges. Let V be an order complete vector sublattice of ba( 31). We prove a decomposition theorem for charges in V which implies the following decomposition. Let M be the set of all charges X e V+ such that X is an extreme point of the convex set of charges in V which coincide with A on a subalgebra of 3t. Then each charge p e V+ has a unique decomposition p = ¡ix + p2, where ju1 is a countable sum of elements of M and p2 is a charge which is disjoint from each element of M. However, it is shown that this decomposition only depends on a certain property of the set M and a suitable characterization of the band generated by M. For several choices of V the elements of M may be characterized by an approximation property [3, 4]. In case V = ba( 31) and V = ca( 31) we derive characterizations of the second component in the above decomposition, where ca( 2Í) denotes the order complete vector sublattice of ba( 31) consisting of the countably additive functions in ba(3t); elements of ca(3t)+ are referred to as measures. We need some further notation. For a subset M of V let B(M) denote the band generated by M in V and M-1 the set of all p e V disjoint from each X e M, i.e. |ju.| a |\| = 0 for each X e M. For a convex subset C of V let ex C denote the set of all extreme points of C. If v, p e V+ with v < p, then v is called a minorant of p..
2. The decomposition theorem. Let M be a subset of V+ with 0 g M. Define
Nx = Ip e V+: p = Y, Xn,Xn e M for every n e N > and N2 = {p e V+ : p admits no nonzero minorant in M}. Lemma 1. Each n e V+ has a decomposition p = px + p2, where px e Nx and p2 e N2. Received by the editors May 25, 1984 and, in revised form, February 1, 1985. 1980 Mathematics Subject Classification. Primary 28A10. Key words and phrases. measures.
Bands generated
by a set of charges, extreme extensions
©
1986
American
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Proof.
HARALD LUSCHGY
Let u e V+ and consider the nonempty set
H={feMR:
£/(*)
(ii). The inclusion (M±) + e N2 is obvious. Conversely, let p e N2 and X e M. By (i) X A p e Nx holds, i.e. X A p = E"=1A„ with Xn e M for every n e N. Since A„ < À A ju,< /i, we obtain Xn = 0 for every n e N and so À A ju = 0.
This yields fieM1. (ii) => (iii). Clearly B(M) +d A^ holds. Conversely, let p e B(Af)+. By Lemma 1 and (ii) ju has a decomposition ¡x = px + p2, where px e Nx and p2 e (M±)+. Since fi2 < ¡x, we obtain ju2 g B(M), hence ju2 = 0. This yields p e Nx. (iii) =» (i). Obvious. Now one obtains from the Riesz decomposition theorem (cf. [1, 1.5.10]) the following decomposition. Theorem. Assume that (P) is satisfied. Then each p e V + has a unique decomposition p = px + p2, where Px G ^i and p2 e N2.
Remark. In case V = ca( 31), where 31 is a a-algebra on ß, consider the following property of M which is stronger than (P):
(P')
p G ca(3I) + ,A G M, ¡i «: a imply ju g M.
Then, under (P'), for ft G ca(3l)+ there exists a countable partition {An: n e I U {0}}, / c N of Q in 31, such that for the components of a in the above decomposition px = ¿Z„erp(An n • ) withju(^n n • ) G M for every n e Iandp2 = p(A0 D ■)
holds. To see this let ® = {A e 3t:ju(yl) > 0,n(A n -)e M). The subsets of ® consisting of pairwise disjoint sets are inductively ordered by inclusion, so by the Kuratowski-Zorn lemma there is a maximal subset {An: n e 1} of this type. Since it is clearly countable, let / c N and A0 = (U„e//ln)c. If A e M,
X < p(A0 n • ), and f denotes a (i(A0 n • )-density of X, then ju(^í0n{/>0} n -)«A.
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DECOMPOSITION OF CHARGES AND MEASURES
By (P') and the maximality p(A0 n -)e N2.
of {An: n e I)
123
this implies X = 0. Thus we obtain
3. Examples. (1) Let 93 be a subalgebra of 3t. For p, e V+ we set £(]u|93) = {v e
V+: v\SQ= ju158}.Let M = {XeV+:Xe
ex£(A|93)}.
Then 0 g M and (P) is satisfied. In fact, if ¡i e V+ is a minorant of X e M and
a = (ii[ + p2)/2, where /i, g £"(^i|39), / = 1, 2, then X = (px + X —p + fi2 + X —p)/2
and /t, + À —/x g £(À|33), ; = 1, 2. It follows X = px + X —p = p,2 + X —p, thus /Xj = ii2. This implies p e M. Therefore the Theorem applies. In order to characterize M and the corresponding set N2 we need the following information. For ¡i e ba(3I)+ define the R^-valued vector charge p on 31 by ¡X= (p(B n •))BeS • p is said to be strongly continuous if for each e > 0 there exists a finite partition {Ax,... ,A„) of Í2 in 3t such that/x(^,) < ejû(n) for every i; ¡X is said to be strongly nonatomic if for each A e 31 and a G [0,1] there exists Ax e 31 such that Ax c A and p(Ax) = ap(A). A charge p on 31 is called 93-approximable if 93 is dense in 31 in the topology of the semimetric (A, B) >-*¡x(AaB). Lemma 3. Let ¡i be a charge on 31. Consider the following conditions: (i) ii has no nonzero '8-approximable minorant in ba(3i)+. (ii) p(A n • ) is not '¡ö-approximable for every A G 31 with p(A) > 0. (iii) jû is a Liapunov vector charge, i.e. {fi(C): C e 3t, C c A} is a convex subset of
R* for every A e 31. (iv) ß is strongly nonatomic. (v) jû,is strongly continuous. Then (i) => (ii) and (iii) => (iv) => (v) => (i) are valid. If, in addition, a-algebra and p a measure, then these conditions are all equivalent.
31 is a
Proof, (i) => (ii) and (iii) =» (iv) are obvious and (iv) =» (v) may be proved similar to the scalar case. (v) => (i). Let v g ba( 3Í ) + be 23-approximable, v < p and let e > 0. By assumption there is a partition {Ax,... ,An) of Í2 in 31 such that p(A¡ ní)< ep(B) for every i and B e 93 and for each / there exists B¡ e 93 such that v(A¡AB¡) < e/«2. Then
"((Û^l
U ¿ p(A,aB,)< e/n
and hence, v(Q.) < v(\J"=1Bi) + e/n. Furthermore, j we have
for each pair of distinct indices i,
p(B¡ n Bj) < v(ACin B¡) + p(^; n 5,.) < 2e/«2.
For the partition {CX,...,C„} of U"=1B, in 93 defined by C, = Bx and CA= Bk \ UfZx% for A:= 2,..., n this yields A—1
w(Bk\Ck)
2. Thus we obtain v(Bj) < ep(CA + 3e/n for every i and so *>(ß)
0 is arbitrary, v = 0 holds. (ii) (iii). Now suppose that 3Í is a a-algebra and p a measure on 3t. According to
[2, Corollary], (iii) holds if and only if {1B: B e 93} is a thin subset of Lx(p) in the sense of [2] and by the Hahn-Banach theorem this is equivalent to the fact that S( 93) is not a norm-dense subset of Lx(p(A n • )) for every A e 31 with p(A) > 0, where 5(93) is the vector space of all 93-simple real functions. To prove that the latter property is equivalent to (ii) we have to show that a measure v on 31 is 93-approximable if and only if 5(93) is dense in Lx(v). The "only if part is obvious. To prove the "if" part let A e 31 and e > 0. There is a sequence in 5(93) which converges to 1A in Lx(v). Since Lx(v\^80) is a Banach space, where 930 denotes the a-algebra generated
by 93, there is a function/g Lx(p\%0) such that f\lA - f\dv = 0. For B0 = {/= 1} we obtain B0 e 930 and v(AaB0) = 0. If we choose Bx e 93 with v(B0aBx) < e, then v(AaBx) < e holds. Part (a) of the following lemma is a generalization of [1, Proposition 5.3.7]. Lemma 4. Assume that 93 is a finite subalgebra of 31. (a) Let % be a a-algebra, % an algebra generating 3Í, and p. a measure on 31. Then p is strongly continuous if and only ifp\(tg'is strongly continuous. (b) Let p be a charge on 3Í. Then juthas no nonzero Së-approximable minorant in ba( 3t ) + // and only if p is strongly continuous.
Proof, (a) We have to prove the "only if" part. We may assume ju(ß) > 0. Let e > 0. Since jû is strongly continuous, there is a partition {Ax,... ,An} of ß in 3Í such
that p(AA < e£(ß)/2 for every i. Let 8 = min{eju(.ß)/2: B e 93, p(B) > 0}. Then 8 > 0. For each i there exists C, g fé'such that p(A¡ACj) < 8/n. Then
mMLKJU ¿mUac,)
