UPPER ENVELOPES OF INNER PREMEASURES

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Ann. Inst. Fourier, Grenoble

50, 2 (special Cinquantenaire) (2000), 401-422

UPPER ENVELOPES OF INNER PREMEASURES

by Heinz KONIG

In the recent book [18] on measure and integration (cited as MI) and in subsequent papers [19] - [24] the present author attempted to restructure the domain of their basic extension and representation procedures, and to develop the implications on various issues in measure and integration and beyond. There were essential connections with the work [7] [8] of Gustave Choquet. Thus MI Section 10 obtained an extended version of his capacitability theorem. The representation theories in MI chapter V and [22] [24] were based on the Choquet integral introduced in [7] Section 48, in form of the so-called horizontal integral of MI Section 11, while [24]

Section 1 obtained a comprehensive version of his fundamental theorem [7] 54.1. The present paper wants to resume another theme initiated in [7]

Section 53.7, that is the representation of certain non-additive set functions and functionals as upper envelopes of appropriate measures.

We quote the definitive result due to Tops0e [28] Section 8 Theorem 2, subsequent to papers of Strassen [26], Dellacherie [9], Anger [3], Fuglede [12], and Huber-Strassen [15]. See also Anger [4] [5], Dellacherie [10], and Tops0e [29].

THEOREM. — Let X be a Hausdorff topological space with the ob- vious set systems Comp(X) and Op(X). Assume that the set function (3 : Comp(X) —> [0,oo[ is isotone with (3(0) = 0 and submodular, and continuous from above in the sense that

for A € Comp(X) and e > 0 there exists U € Op(X) with A C U such that f3{K) < (3(A) + e for all K e Comp(X) with K C U.

Keywords : Submodular isotone set functions - Inner premeasures - Supportive pro- perties.

Math classification : 28A10 - 28A12 - 28A25 - 28C05 - 28C15 - 46A22.

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Then for each A 6 Comp(X) there exists a Radon premeasure (p : Comp(X) -^ [0,oo[ such that (p ^ /3 and (^(A) = /?(A). Likewise if sup/? < oo then there exists a Radon premeasure (p : Comp(X) —> [0,oo[

such that ^p ^ f3 and supy? = sup (3.

Then Adamski [1] transferred the theorem to the frame of abstract measures. He assumed certain pairs of lattices © and 1 of subsets in an abstract set X to take the place of Comp(X) and Op(X). With new ideas he was able to obtain fortified results. Now the present paper wants to show that the frame of MI leads to even more fortified and simpler forms of the results. One of the simplifications is that Adamski [1] assumed G to be stable under countable intersections and the initial set function f3 : © —>• [0, oo [ to be a continuous at 0, which at once leads to the level of measures, whereas we shall see that the adequate level is the so-called finitely additive one, that is the level of contents. Also we shall compare our basic result with the main theorem of MI Section 18, at that place called the extended Henry-Lembcke-Bachman-Sultan-Lipecki-Adamski theorem, this time after Adamski [2]. There will be some remarkable consequences and examples.

Besides measure and integration the basic methodical device for the present area is an appropriate Hahn-Banach type theorem (or an equivalent assertion), as it became clear in particular from Tops0e [28] Section 8. Now the relevant Hahn-Banach theorems in the literature are of delicate nature and proof; see the work of Anger-Lembcke [6] referred to in Tops0e [28], or Fuchssteiner-Konig [II], and in particular the theorem of Rode [25] [17].

Thus it is perhaps not superfluous to present a certain special case of the Rode theorem which suffices for the present purpose and has a short and simple proof.

Basic notions and notations. — We adopt the terms of MI but shall recall the less familiar ones. Let X be a nonvoid set. For S C X the complement is denoted 5", and for a set system © in X we write 6-L := {S^/ : S C ©}. For set systems 0 and⁽! in X we form the transporter ©T? := {A C X : A D S e <! V5 6 ©}. For a set function 9 : VW —^ [°^ °°] with Q(0) = 0 we recall the Caratheodory class

€(Q) := {A C X : Q(M) = 9(M H A) + 6(M H A') for all M C X}.

€(0) turns out to be an algebra, and 9|(£(G) to be modular.

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UPPER ENVELOPES OF INNER PREMEASURES 403

The extension and representation theories in MI and in the subsequent [22] [23] [24] come in three simultaneous versions. They are marked • = -A-CTT, where * is to be read as finite^ a as sequential or countable, and r as nonsequential or arbitrary (or as the respective adverbs). Moreover the theories come in parallel outer and inner versions, that means in versions which are based on outer and inner regularity. The extension theories for set functions are summarized in [22] Section 1 and [23] Section 1. The present paper will concentrate on the inner * version, but it is of course vital to note the consequences for the inner or versions which result from routine combinations with the means of MI.

We recall the relevant envelope formations: Let © be a lattice of subsets of X with 0 € ©, and y : G —>• [O.oo] be an isotone set function with ^(0) = 0. One defines the outer and inner -k envelopes (^, ^ : q3(X) -^ [0, oo] of ^ to be

(^-(A) = inf{(^(5') : S € 6 with S D A},

^(A) = sup{y?(6') : S eG with S C A}.

The inner versions require that ip : G —^ [0,oo[ be finite. We consider the inner ^ version: One defines an inner -A- extension of (p to be a content a : 21 —>• [0, oo] on a ring 21 D 0 which extends ^p and is inner regular 0.

One defines (p to be an inner -A- premeasure iff it admits inner * extensions.

The subsequent inner * theorem characterizes those (/? which are inner ^ premeasures, and then describes all inner * extensions of ip. The theorem is in terms of y^; its essence can be found earlier in Tops0e [27] Section 4.

INNER -A- THEOREM.— Assume that ( / ? : © — > [0,oo[ is isotone with (^(0) = 0. Then the following are equivalent:

1) (p is an inner -k premeasure.

2) (^|€((^.) is an inner -*- extension of (p.

3) (^*|^(^) is an extension of (p.

4) (p is supermodular, and inner -A- tight in the sense that

^p{B) ^ (/?(A) + (p^B \ A) for all A C B in G.

In this case all inner * extensions of(p are restrictions of(p^\€((p^). Moreover 6T£(^) C £(^).

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1. The Hahn-Banach type theorems.

The first theorem is an obvious special case of the Hahn-Banach theorem due to Rode [25] (with Zusatz p. 480). The point here is its short and simple proof.

THEOREM 1.1.— Assume that the nonvoid set E carries an asso- ciative and commutative addition + and an order relation ^, which are compatible in the sense that u ^ v = > u - { - x ^ v - ^ - x for all u, v^ x € E. Let

Q : E —>} —oo, oo] be subadditive and isotone, P : E —>] —oo, oo] be super additive and isotone,

and P ^ Q. Then there exists an additive and isotone function f : E —>

] -oo, oo] such that P ^ f ^ Q.

Proof. — Let Q : E —^]—oo, oo] be subadditive and isotone, and define M(Q) to consist of all superadditive and isotone functions / : E —»]—oo, oo]

with / ^ Q.

0) M(Q) is upward inductive in the pointwise order. In fact, if H C M(Q) is nonvoid and upward directed then the pointwise supremum / := svip^ffh is in M(Q). Thus each P e M(Q) has maximal members / € M(Q) with P ^ /. It remains to show that each maximal / € M(Q) must be subadditive and hence additive.

1) We claim that f{nx) = Tif(x) for x C E and n G N, where of course nx := x-\" • ' - } - x (n terms). First of all f(nx) ^ nf(x) and Q(nx) ^ nQ(x).

For fixed n C N now consider F : F{x) = ^f(nx) for x e E. Then F e M(Q) and F ^ /, and hence F = / as claimed.

2) We claim that f{x 4- a) ^ f(x) + Q(a) for x, a 6 E. To see this fix a E E with Q(a) < oo, and define F : E —^—oo, oo] to be

F(x) = sup{/(a; + no) - nQ{a) : n ^ 0} for x G E,

with the obvious role of n = 0. Then -F is isotone and F ^ f. We have F ^ Q because

f(x + no) - nQ(a) ^ Q(x + no) - nQ{d) ^ Q{x} + Q(na) - nQ(a) ^ Q(x),

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and F is superadditive because for u, v e E and m, n ^ 0 we have F(zA + v) ^ f((u + v) + (m + n)a) - (m + n)Q(a)

= /((z( + ma) -h (i; + na)) — (m + n)Q(a)

^ (/(zA + ma) - mQ(a)) + (/(v + no) - nQ(a)).

Thus F C M(Q) and F ^ f, and hence F = f. For n = 1 the assertion follows.

3) We claim that f(x + a) ^ /(.r) + /(a) for x,a e E, which will complete the proof. To see this fix a € F with /(a) < oo, and define F : E ->} -00,00] to be

F(x) = sup{/(a* + no) — nf(a) : n ^ 1}

== lim [f(x + no) — nf(a)] for x € E\

n—>oo \ /

note that the expression in the last brackets increases with n ^ 1. Then F is isotone and F ^ /, and F ^ Q from 1)2). As in 2) one proves that F is superadditive. Thus F C M(Q) and F ^ /, and hence F = /. For n = 1 the assertion follows. D

We combine the above result with the sub/superadditivity theorem for the Choquet integral in the version MI 11.11 to obtain our basic device.

This is a known theorem too; see Kindler [16] Section 5 Example 1 and MI 11.24. The present proof has been sketched in MI 11.14.

THEOREM 1.2.— Let © be a lattice of subsets in X with 0 e 6.

Assume that

(3 : G —» [0, oo] is isotone with (3(0) = 0 and submodular, a : © —f [0, oo] is isotone with 0(0) = 0 and supermodular, and a ^ f3. Then there exists an isotone and modular set function y : G —> [0, oo] such that a ^ ^ ^ /3.

Proof.— Let E consist of the finite linear combinations of the characteristic functions \s of the S € 6 with coefficients ^ 0, that is E = S(©) in the sense of MI Section 11, equipped with pointwise addition + and order relation ^. Define Q^P : E —> [0,oo] to be Q(u) = -f- udf3 and P(u) = •f- uda for u € £', where f denotes the horizontal integral of MI Section 11. After MI 11.11 then Q and P are as required in 1.1. Hence there exists an additive and isotone functional / : E —^ [0, oo] such that

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P ^ f ^ Q' Define y : © -^ [0, oo] to be (p(S) = /(^) for S C ©. Then (^

is isotone and modular and fulfils a ^ y? ^ /3. D We conclude the section with a short supplement which will not be needed in the sequel. The next assertion is the central result of Horn-Tarski [14] specialized to lattices (but with the value oo admitted).

THEOREM 1.3.— Let © be a lattice in X with 0 € ©. Then each isotone and modular set function ^p : 6 —> [0, oo] with y?(0) = 0 can be extended to a content (f): ^(X) —» [0, oo] with 4>(X) = sup y?.

Proof. — The extension ^ : © U {X} —> [0, oo] of ^, in case X ^ 6 with 'ff(X) = supy?, retains the assumptions. The assertion then follows from 1.2 applied to the pair ^ ^ i^*. D Example 1.4. — Let X = [0,1]. Define © to consist of 0 and of the [0,t[ with 0 < ^ 1, and ^ : © -> [0,oo[ to be y?(0) = ^([0,^[) = 0 for 0 < t < 1 and <^([0, ![)=!. Thus 6 is totally ordered under the inclusion C a.nd hence a lattice, and y? is isotone and modular. Let (f) : ^(X) —> [0, oo]

be as in 1.3, and '9 := 0[Comp(X). Then 'ff : Comp(X) —> [0, oo[ is isotone and modular with ^(0) = 0, but not inner -A- tight, because ^([0,1]) = 1 and ?9({1}) == i9^([0,l[) = 0. Thus ^ is not an inner -A- premeasure, that is not a Radon premeasure. We note that we have not seen such examples in the literature so far. Their existence has been overlooked for example in [27] p.4 1.14-24.

2. Some preliminaries.

The present section assumes a pair of lattices © and T in X with 0 € ©.I. An illustrative example is © = Comp(X) and 1 = Op(X) in a Hausdorff topological space X, as discussed in the introduction and resumed in 2.7 below. After some simple remarks we consider certain properties of separation.

Remark 2.1. — For isotone set functions y?: © —> [0, oo] and ^ : T —>

[0, oo] with ^(0) = ^(0) = 0 we have ^[T ^ ^ <^> </? ^ ^|6.

Proof. — Both relations mean that all pairs 5 G © and T G T with 5 C T fulfil ^(6') ^ ^(T). D

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UPPER ENVELOPES OP INNER PREMEASURES 407

Remark 2.2. — For an isotone </?: © —> [0, oo] with ^(0) = 0 we have (p ^ (^ll)*!®. Moreover <^ = (^ll)*!^ iff ^ = '0*|6 for some isotone -0 : ? -^ [0, oo] with -0(0) = 0.

Proof. — The first assertion is clear from the definitions. We prove the second one. =^) ^ := ^IT is as required. ^=) We have y^l? = (^©^IT ^ ^ as in the first assertion, and hence (^^{(S ^ ^\G = y?.

D

Remark 2.3. — Assume that the isotone and submodular ^ : Ti —>

[0,oo] with ^(0) = 0 fulfils ^ := -0*|6 < oo, so that i? : © -> [0,oo[ is isotone with ^(0) = 0. If T C (©T©)-L then '9 is inner ^ tight.

Proof. — To be shown is ^(B) ^ ^(A) + ^(B \ A) for A C B in ©.

For fixed c > ^(A) there exists T e ^ with T D A and ^(T) < c. Then T' H B e G with r' n B c A' n B = B \ A. Therefore

^(B) = '0*(B) ^ ^{B U T) ^ ^(T) + ^(T' n B)

= ^(T) + ^(r' n B) < c + ^(^ \ A).

The assertion follows. D We turn to the announced properties of separation. Let as before © and T be lattices in X with 0 € 6.T. We say as usual that T separates G iff for each disjoint pair A, B € © there exists a disjoint pair [7, V G 1 with A C L7' and B C V. We need two further properties. On the one hand we define a pair A, B C X to be separated T iff for each M e 1 with A n B c M there exists a pair U, V € 1 with A C ^7 and B C V such that UnV C M. On the other hand we define a pair A, B C X to be coseparated

© iff for each M e © with M C A U B there exists a pair P, Q € © with P C A and Q C B such that M C P U Q. These two properties came up in MI 4.2 and MI 6.4. We recall the consequences obtained at these places.

Remark 2.4.— 1) Let (p : © —> [0,oo] be isotone and submodular with y?(0) = 0. If the pair A,B C X is coseparated © then

^(A U B) + <^(A n B) ^ <^(A) + ^(B).

2) Let ^ : T —^ [0, oo] be isotone and supermodular with ^(0) = 0. If the pair A, B C X is separated T then

^(A U B) + ^-(A n B) ^ ^-(A) + ^(B).

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Thus each time the semimodular behaviour carries over to the less natural and unexpected '*' envelope. We come to the basic relations which connect the notions defined above.

PROPOSITION 2.5.— Assume that 1 separates ©, and that <! C (6T©)_L. 1) Each pair A, B € T is coseparated ©. 2) Each pair A, B e © is separated T.

Proof. — 1) Fix A, B e <! and M C © with M C A U B. Then A',B' e ©T© and hence M D A',M H B' € ©, and these two sets are disjoint. Hence there exist disjoint U, V € 1 with M D A' C U and MHB' C V, that is with M C AUU and M C BUV. Thus P := MW C © and Q :== M H V € © fulfil P C A and Q C B. Moreover U ' U V = X implies that P U Q = M.

2) Fix A, B e 0 and M e T with A D B C M. Then M' e ©T6 and hence A D M',B D M' € ©, and these two sets are disjoint. Hence there exist disjoint P, Q e T with A D M' C P and B D M' C Q, that is with A C M U P and B C M U Q. Thus L ^ M U P e l a n d y ^ M u Q e l are as required. D CONSEQUENCE 2.6.— Assume that T separates ©, and that I C (©T©)J_. 1) Let < / ? : © — > [0,oo] be isotone and submodular with (p(0) = 0. Then ^IT is submodular. 2) Let ^ : T -^ [0,oo] be isotone and supermodular with ^(0) = 0. Then ^*|© is supermodular.

Example 2.7. — Let X be a Hausdorff topological space. Then 6 :=

Comp(X) and T := Op(X) fulfil both⁽! C (0T©)J_ and © C (TTT)!, and T separates 6.

We conclude the section with the overall remark that part of its results can be extended to the common frame • = wr, while others lead to serious problems. Thus the situation is different from [23] part I, but similar to MI chapter VI.

3. The basic results.

As before we assume a pair of lattices 6 and 1 in X with 0 C 6, T.

For an isotone set function (p : 6 —> [0,oo] with y?(0) = 0 we write (p := (^IT)*!^. Thus <f> : © —^ [0,oo] is isotone with (^(0) = 0. The

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above 2.2 says that (p ^ (p and has a simple equivalence for the case (p = (p, in which case ^ is sometimes called semi-regular.

Example 3.1. — In a Hausdorff topological space X let © = Comp(X) and ? = Op(X). Then for an isotone ^p : G —> [0,oo] with y?(0) == 0 the relation (p = (p means that ^p is continuous from above as denned in the theorem quoted in the introduction.

Example 3.2. — . Assume that (5 is upward enclosable T, denned to mean that each member of 6 is contained in some member of T. Define (p : G —> [0, oo[ to be (^(0) = 0 and ^(S) = 1 for nonvoid S € (3. Thus (^ is isotone and submodular. It is immediate that y? = (p.

The next result extends [27] Lemma 2.4 and [1] Lemma 3.1(b)(c).

In its present form it can be considered as an abstract version of classical results in Halmos [13] Sections 53-54.

PROPOSITION 3.3.— Assume that (! C (©T6)-L, and that 1 sepa- rates 6. For an isotone (p : @ —> [0, oo[ with y?(0) = 0 then

if^p is modular and (p < oo: (p is an inner -A- premeasure.

Moreover

if(p is submodular: (p = (p ==^> ^p is inner ^ tight and (p < oo, if^p is supermodular: (p = (p <^== (p is inner ^ tight and (p < oo;

both times the converse need not be true. In particular

if(p is modular: (p = (p ^==> (p is inner -*- tight and (p < oo.

Proof. — We have (p = ip*\G for '0 := ^IT, where ^ : T —> [0, oo] is isotone with '0(0) = 0.

1) Let (p be supermodular. Then ip is supermodular, and hence (p is supermodular by 2.6.2).

2) Let y? be submodular. Then ip is submodular by 2.6.1), and hence (p is submodular. If moreover (p < oo, then (p is inner * tight by 2.3.

3) The first assertion follows from 1)2), and the second one from 2). We turn to the third assertion. The converses will be dealt with in 3.4 below.

4) Let (p be supermodular and inner -k tight, that is an inner -A- premeasure, with (p < oo. To be shown is (p ^ ^p. We fix S G 6, and then T e ? with T D S and ^(T) < oo. For fixed e > 0 we take K e 6 with K C T \ S and ^p(K) > ^(T \ S) - e. In view of © C €(^) and

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T C (©T©)-L C C(^) the last relation can be written (p(S) > ^(T\K)-£.

Now for the two disjoint S^ K € 6 there exist disjoint U^ V C ^ with S C U and Jf C V; we can of course assume that L^, V C T. Then

£7 C T n V C T H ^' = r \ K and hence

<^) ^ ^(£/) ^ (^(T \ K) < ^(S) + ^.

The assertion follows. D Example 3.4. — There are simple counterexamples which disprove the two converse assertions.

1) Assume that G contains the one-point subsets of X, and fix a subset K e © which is not in T. Define (p : 6 -^ [0,oo[ to be (^(S') = 0 for S C K and <^(5') == 1 for S (f. K. Then y? is isotone and submodular, and one verifies that (p is inner -A- tight and (p ^ 1 < oo. But ^(-^) = 0 and (p(K) = 1, so that ip ^ (p.

2) In a set X of more than one element let © consist of the finite subsets and T D ©; then 6 and 1 are as required. Define (p : © —> [0, oo[

to be (p(S) = (#(5))2 for 5' € 6. Then (^ is isotone with (p(0) = 0, and

^ = (p < oo in view of (! 3 ©. For A,B e G with #(A) = m, #(B) = n and #(A D B) = p we have

<^(A U B) + (^(A n B) - (p(A) - (p(B) = (m + n - p)2 + p2 - m2 - n2

=2(m-p)(n-p).

Thus (p is supermodular. But since © is a ring the case p = 0 shows that (/? is not inner -A- tight.

For the remainder of the section we fix an isotone set function (3 : 6 —> [0,oo[ with (3(0) = 0. We define M{/3) to consist of the isotone and supermodular set functions ^p : 6 —> [0, oo[ with (p ^ f3.

Remark 3.5. — 1) M(/3) is upward inductive in the setwise order.

2) If (3 is submodular then each maximal member ofM{f3) is modular.

Proof. — 1) If H C M(/3) is nonvoid and upward directed then the setwise supremum i9 := supy,^ y? is in M(/?).

2) Follows from the Hahn-Banach Theorem 1.2. D The next theorem and the subsequent consequence are the basic results in the present context.

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THEOREM 3.6.— Assume that T C (©T©)J_, and that T separates (3. Let (3 be submodular and (3 = (3. Then each maximal member (p G M(/3) is an inner ^ premeasure and fulfils ^p = (p.

Proof. — i) We see from 3.5.2) that (p is submodular.

ii) ^ := ^l? is supermodular, and hence (p = '0*|6 is supermodular by 2.6.2). Moreover (p ^ (p ^ f3 = f3. Therefore (p = (p.

iii) From i)ii) combined with 3.3 we see that (p is inner -A- tight and hence an inner -*- premeasure. D CONSEQUENCE 3.7.— Assume thatc! C (6T6)_L, and that 1 sepa- rates G. Let (3 be submodular and ( 3 = ( 3 . I f y j l C © i s a lattice such that /3|9Jt is modular then there exists an inner -*- premeasure ^p : G —^ [0,oo[

with y ^ ( 3 and ^|9Jt = (3\W.

Proof.— We can assume that 0 6 9Jt. Then a := (/3|9Jt)^|© is in M(/3). Thus there exists a maximal member (p 6 M(/?) with a ^ (^ ^ /3, that is with (^ ^ /? and ^|W = /?|9Jl. D SPECIALIZATION 3.8.— Assume that 1 C (©T©)±, and that 1 separates 6. Let (3 be submodular and (3 = (3. IfWc © is nonvoid and totally ordered under inclusion then there exists an inner -A- premeasure ip : © -^ [0, oo[ with y ^ (3 and ^|9Jt == /3|9Jt.

The case 9DZ = {A} C © in the special situation of Example 3.1 is the first assertion in the theorem of Tops0e [28] quoted in the introduction.

Its second assertion is contained in 3.9 below. The case that 9JI consists of the members of a sequence 5'i C • • • C Sn C • • • in © is in Adamski [1]

Corollary 3.7, in essence under fortified assumptions. This special case has the consequence which follows.

CONSEQUENCE 3.9.— Assume that T C (©T©)_L, and that 1 sepa- rates ©. Let f3 be submodular and f3 = f3. For each M C X there exists an inner -k premeasure ^ : © —> [0, oo[ with (p ^ (3 and <^(M) = /3*(M).

Proof. — Fix a sequence 5'i C • • • C Sn C • • • C M in © such that /?(5n) T ^(M). The result follows from 3.8 applied to 9Jt := {Sn : n 6 N}.

D

So far the basic results in the present context. It is of interest to compare 3.6 with the main theorem MI 18.10 of MI Section 18. We present this result (in the so-called conventional situation of MI) in the

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reformulation which follows. Let as above 6 and 1 be lattices in X with 0 € ©, T and (3 : © -^ [0, oo[ be isotone with (3(0) = 0.

THEOREM 3.10. — Assume that (3 == f3. Then each maximal member (p € M(/3) such that (/^l? is an inner * premeasure is itself an inner -*- premeasure.

It is not obvious that the above 3.10 is equivalent to MI 18.10.

Therefore we shall include the proof of this equivalence. For an isotone and supermodular set function ^ : T —>• [0,oo[ with ^(0) = 0 let -*'('0) as in MI Section 18 consist of the isotone and supermodular ^ : © —> [0, oo[

with y?(0) = 0 such that ^*|T == '0. Then MI 18.4 says that ^(if^) is upward inductive in the setwise order, and MI 18.10 can be formulated as follows:

Assume that © is upward enclosable T. If ^ : ? —^ [0,oo[ is an inner * premeasure, then each maximal member y? : © —> [0, oo[ of^-(^) is an inner -Ar premeasure.

Proof of MI 18.10 =^ 3.10. — Let (3 and (p be as assumed in 3.10. Then first of all f3 < oo implies that © is upward enclosable T. By assumption

^ := (^IT is an inner -A- premeasure, and by definition ^p € *(^). It suffices to show that y? is a maximal member of-A-('0). Thus let i9 6 -*-(^) with (p ^ i9. To be shown is (/? == ^. From ^|T = ip and 2.1 we have ' ^ ^ ' 0 ' * ' | © = = ( ^ ^ / 3 = / ? and hence ^ € M(/?). By assumption y? is a maximal member of M(/3), and hence y? = ^. D Proof of 3.10 => MI 18.10.— Assume that © is upward enclosable

^T, and let '0 and y? be as assumed in MI 18.10. Then f3 := '0*|© is < oo and isotone with (3(0) = 0, and (3 = (3 by 2.2. From ^l? = ^ and 2.1 we have (p ^ ^\6 = (3 and hence (/? 6 M(/?). It remains to show that (p is a maximal member of M(/3). Thus let ^ € M(/?) with (/? ^ ^. To be shown is ^ = ^9. We have ^⁼ ^|^ ^ ^1^ ^ ^1^; and from (3 = ^\6 and 2.1 also /^IT ^ '0. Thus ^IT = -0 and hence t9 e ^(VQ' By assumption y? is a maximal member of -*'('0), and hence (p = ^. D In MI sections 18 and 19 there were numerous applications of MI 18.10. In the present context of 3.10 we shall restrict ourselves to the particular case T := {0, X}, because then each set function ^ : 1 —^ [0, oo[

with ^(0) = 0 is an inner '*' premeasure. In this case (3 = (3 means that f3(S) = sup (3 < oo for all nonvoid S e ©. Thus we fix f3 : © -^ [0, oo[ to be f3(0) = 0 and (3(S) = 1 for S + 0. Then 3.10 says that each maximal member of M(/3) is an inner * premeasure. We emphasize one particular consequence.

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CONSEQUENCE 3.11. — Let G be a lattice in X with 0 € ©. Assume that the nonvoid 97t C V(X) is downward directed and upward enclosable G with 0 ^ QJt. Then there exists an inner * premeasure </? : 6 —^ [0, oo[

with (p ^ 1 and ^*|9Jt = 1.

Proof. — Define a : © —> [0, oo[ to be a(S) = 0 when there is no M € 9Jt with M C S, and a(S') = 1 otherwise. It is clear that a e M(/3) for the above /3 : G —> [0, oo[. Thus each maximal member (p € M(/?) with a ^ y? is as required. D Remark 3.12. — In 3.10 it cannot be achieved as in 3.6 that y? == (^.

In fact, in the particular case 3.11 this would mean that y?(5) = 1 for all nonvoid S € ©, so that y? could not be modular whenever there are nonvoid A, B C @ with A D B == 0.

It is remarkable that the above 3.11 can also be obtained from the basic Theorem 3.6, this time via 1 = (6T@)_L and with the same proof.

However, one needs the additional assumption that (6T©)J- separates 6, but in return one can achieve that </? = (p.

We conclude with two further consequences which underline the wealth of inner * premeasures. The results are in the terms of MI Sections 6 and 11.

PROPOSITION 3.13. — Let G be a lattice in X with 0 € 6 which for some • = err is not • compact. Then there exists an inner -A- premeasure (p : G —> [0, oo [ which is not • continuous at 0.

Proof.— The assumption means that there exists a nonvoid and downward directed set system 9Jt C 6 of type • such that 0 ^ 9?l but CT [ 0. Thus the assertion follows from 3.11. D PROPOSITION 3.14.— Let G be a lattice in X with 0 € ©. Assume that f : X —^ [0, oo[ with sup / = 1 has 9JI := {[/ ^ t] : 0 < t < 1} upward enclosable ©. Then there exists an inner * premeasure y?: © —> [0, oo[ with (p ^ 1 such that -f- fd^p* = 1.

Proof. — Follows from 3.11 applied to W. D Example 3.15.— The last assertion becomes false when instead of -f- fd(p* = 1 one requires that •f- fd^ = 1. As an example take X = [0,1]

and 6 = Comp(X), and / : f(x) = x for 0 ^ x < 1 and /(I) = 0.

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4. Extension of the basic consequence.

The present section combines the basic consequence 3.7 with the ideas of Adamski [1] Lemma 3.2(b)(c) in order to obtain an extended version.

We use the ideas in question as expressed in the next two lemmata. The initial part of the section assumes a lattice 0 in X with 0 e ©.

LEMMA 4.1.— Let ^p : 6 —>• [0,oo[ be an inner ^ premeasure. For fixed 9Jt C 6 non void and upward directed define a : © —^ [0, oo[ to be

a(S) = sup{<^(6' H M) : M e W} for S e 6.

Then a is an inner -k premeasure. Note that a ^ y?, and a(S) = ^p(S) for S C 0 upward enclosable 9Jt; therefore

a{S) = sup{a(5' H M) : M e 9Jt} for S 6 ©.

Proof. — It is obvious that a is isotone and as claimed in the last sentence, and a routine verification that a is supermodular. Now let A C B in 6. For M € W then

y(B H M) ^ (p(A n M) 4- sup{^(5) : S € © with S C (B H M) \ (A H M)}

= <^(A n M) 4- sup{y?(5' n M) : S € © with ^ C 5 \ A}

^ a(A) + sup{a(S') : S € © with 6' C B \ A},

so that a{B) ^ a(A) + a^(B \ A). D LEMMA 4.2.— Let (/? : © —^ [0,oo[ be isotone and submodular with y?(0) = 0. For fixed 9Jt C © nonvoid and upward directed define a : © —> [0, oo[ to be

a(S) = inf{^(5 U M) - <^(M) : M e 9Jt} for 5 € 6.

0) For fixed S € © the set function M i-> ^(5 U M) - (^(M) on 6 is antitone.

1) a is isotone and submodular with a ^ y?.

2) Jn case c := sup{<^?(M) : M € 9DT} < oo we have 0(5') = sup{^(5 U M) : M 6 9Jt} - c for S G ©.

3) We have a^(T) - 0(6') ^ ^(T) - (^(S') for all S C © and T D 6.

Thus if? is a lattice in X with 0 e T such that y? = (p then a = a as well.

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Proof. — 0) For P C Q in © we have

^(S U Q) - ^(Q) = <p((S U P) U Q) - ^(Q)

^ ^{S UP)- ^((5 u P) n Q) ^ ^{s UP)- (^(P).

1) It is obvious that a is isotone with a ^ y?. To see that a is submodular let A,B C ©. For P,Q C 97t and M e 9Jt with P,Q C M then 0) implies that

a(A U B) + a(A H B) ^ ((/?((A U B) U M) - y?(M)) + (^((A H B) U M) - y(M))

^ (y?(A U M) - y(M)) + (^{B U M) - <^(M))

^ (^(A UP)- (^(P)) + (^(B U Q) - ^(Q)).

The assertion follows.

2) To prove ^ let P, Q C 9Jt and M e 9JI with P, Q C M. From 0) then (^ U P) ^ (^(5 U M) = <^(M) + (^(5 U M) - (^(M))

^ c + ((^(5' U M) - y?(M)) ^ c + (y?(5 U Q) - 99(M)).

The assertion follows. The direction ^ is an immediate consequence of (^(M) + a{S) ^ y(S U M) for M € SJt.

3) Fix S € 6 and T D 6'. For ^ G © with ^ C T and M e 9Jt we have (^ U M) - (^(5' U M) ^ (^((5 U K) U (5 U M)) - ^(S U M)

^ ^(6' U ^T) - (^((5' U K) n (6' U M))

^^SuK)-y(S)^

^p(K U M) - (^(M) ^ (^(5 U M) - y?(M)) 4- ^(5' U K) - y(S),

a{K) ^a(S) + ^(S U K) - ^{S) ^ a(S) + ^(T) - ^(S).

It follows that a^(T) ^ a(S') + ^(T) - (p(S). D Next we need some further terms. For 9Jt C ^P(X) nonvoid and upward directed we define ©(9Jt) C © to consist of the S G © which are upward enclosable 9JI. Thus ©(9Jt) is a lattice with 0 e ©(971). For isotone set functions {3 : G —^ [0,oo[ with f3(0) = 0 and (^ : © —^ [0,oo[

with (p(0) = 0 we say that (p supports f3 at W iff (p ^ /? and ^|9Jl = ^|2Jl.

We say that y? minisupports /3 at 9JI iff in addition

y?(5) = sup{(^(5 n H ) : H e ©(mi)} for 5 € ©;

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this means that (p achieves to support (3 at SO? with minimum expenditure.

We note an immediate consequence of 4.1, and then deduce from 4.2 the main device for the sequel.

Remark 4.3. — Let (p : © —> [0, oo[ be an inner -A- premeasure which supports (3 at 9JI. Define a : © -> [0, oo[ to be

a(S) = snp{^(S n H ) : H e G(W)} for S € ©.

Then a is an inner * premeasure which minisupports (3 at 9Jt.

LEMMA 4.4. — Assume that 9Jt, 9^ C ^(X) are nonvoid and upward directed with M C N for all M e 9Jt and N e 9^. Let f3 : © -> [0, oo[ be isotone and submodular with f3(0) = 0, and define a : © —^ [0,oo[ as in 4.2 to be

a(S) = mf{(3{S U H) - f3(H) : H e 6(9Jt)} for S e ©.

Assume that

^9 : 6 —^ [0, oo [ is an inner ^ premeasure which minisupports /3 at 9Jt, (p : © —> [0, oo [ is an timer * premeasure which minisupports a at 9T.

Then 0 := i9 + (p is an inner * premeasure which minisupports (3 at 9Jt U 9T.

Proof. — 0) It is clear that 0^ = ^ + ^* on ^(X) and that 0 is an inner ^ premeasure.

1) We claim that 6 ^ (3. For S C 6 and H e ©(9JI) we have

^{S n H) + <^(5') ^ ^(5 n AT) + 0(6')

^ /?(6 n H) + /?(6 U H) - (3(H) ^ /?(6), and hence in fact 0(S) = ^(5) + (^(S') ^ l3(S).

2) From the definition we have i9 ^ ^; on (3(9Jt) moreover a = 0 and hence ^ = 0, so that ^ = 0.

3) For 5' e © we have

sup{<9(s' n R) = ^(s n R) + ^(5' n R): R € ©(mi u ^n)}

= sup{^(5' n H) + ^ n ^): H e ©(mi) and R e ©(^n)}

=^)+^(5)=0(5).

4) We claim that 6^(M) = ^(M) for M G 97t. In fact, from 2) we have

^(M)=^(M)=^(M).

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= sup{^(M) : M e mi} ^ ^(AQ,

so that the assertion is clear when c = oo. Thus we can assume that c < oo.

Then 4.2.2) furnishes

0^(N) =^{N) + ^(N) = ^(N) + a^N)

because i^(7V) ^ c as shown above. The assertion follows. D For the remainder of the section we fix a pair of lattices © and 1 in X with 0 C ©,1 and an isotone set function /3 : © —^ [0, oo[ with f3(0) = 0.

THEOREM 4.5.— Assume that 1 C (6T6)J_, and that 1 separates

©. Let (3 be submodular and f3 = J3. If TO c ^(-X) is nonvoid and well-ordered under inclusion then there exists an inner -A- premeasure

Proof. — We can assume that 0 C TO.

1) A nonvoid subsystem ^ C TO is called initial iff A C B for A e TO and 5 € ^ implies that A € y. Thus l.i) {0} C TO is initial, l.ii) If V, Q C TO are initial then either q? C Q or Q C ^P. In fact, if Q (Z' q3 and B € Q but B ^ ^, then an A e ^ cannot fulfil B C A and hence must fulfil A C B, so that A € Q. Thus qj C 0.

2) Define S to consist of all pairs (^P^), where ^ C TO is an initial subsystem and $ : © —> [0, oo[ is an inner -*- premeasure which minisupports (3 at y. S is nonvoid, because the pair (^P,^) with ^ = {0} and $ = 0 is in E. For (q^ $), (Q, 77) c E we define (^, Q C (Q, 77) iff <p c Q, and ^ 77 on © and ^ = rj on ©(^P). This is of course an order relation on E.

3) We claim that E is upward inductive in C. To see this let A c E be nonvoid and totally ordered under C.

3.i) First of all ^ := Ucp^eA^P C TO is an initial subsystem. Define

^ := sup((p^^$, so that ^ : © -^ [0,oo[ with 'Q ^ f3. It is a routine verification that i^ = sup^^^ on ^(X) and that i9 is an inner ^ premeasure.

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3.ii) For V € 9J there exists (⁽?, 0 € A such that V e⁽? and hence

^(y)=A(v).

3.iv) All this proves that (^J,^) € S.

3.v) It remains to show that (^, $) C (^J, 19) for each (^ ^) € A. It is clear that ^ C V and $ ^ i9 on ©. On 6(^) we have

for (0,77) C A with (Q^rj) E (y^): 77 ^, for (0,77) € A with (qU) C (Q^): 77 = ^;

therefore 77 ^ $ for all (Q, 77) € A, so that ^ ^ ^ and hence i9 = ^. Thus in fact(MC(9J^).

4) Now let (9J, 79) be a maximal member of S. We have to prove that

^J == 3Jf. If not, then there is a smallest M C 9Jt which is not in ^O. Then 4.i) V C M for all V 6 2J, because 2J C 9Jt is an initial subsystem.

4.ii) By the choice of M the subsystem QJ U {M} C 9DZ is initial as well.

4.iii) We shall invoke 4.4 for the set systems 9J and {M} and for ft\

the respective a : G —> [0, oo[ is submodular and fulfils a = a from 4.2.1)3).

which minisupports a at {M}. Now 4.4 asserts that 6 := 79 + ^ is an inner -k premeasure which minisupports f3 at VU {M}. Combined with 4.ii) this says that (QJU {M},0) e S. Moreover <^^) F (21 U {M},(9), because on 6(2J) one has a = 0 and hence 0 = i9. This contradicts the assumption that (9J, 79) be a maximal member of S. The proof is complete. D

The final result then reads as follows.

W C G is a lattice such that f3\yjl is modular, and 71 C y{X) is nonvoid and well-ordered under inclusion,

such that M C N for all M C 9JI and N e 9T. Then there exists an inner ^ premeasure (p : 6 —> [0, oo[ which minisupports (3 at 9Jt U 71.

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Proof. — After the model of part 4.iii) in the last proof we shall invoke 4.4 for 9Jt and 9^ and for ^; the respective a : G —> [0,oo[ is submodular and fulfils a = a from 4.2.1)3). From 3.7 and 4.3 we obtain an inner ^ premeasure ^ : © —^ [0,oo[ which minisupports f3 at mi, and from 4.5 we obtain an inner ^ premeasure (p : © -^ [0,oo[ which minisupports a at 9^. Then 4.4 asserts that 0 := i9 + ^p is an inner * premeasure which minisupports /? at 9Jt U ^Tl. D The result obtained in Adamski [1] Proposition 3.8 is for the case that 9Jt consists of the members of a sequence Si C • • • C Sn C • • • in G and that 9T = {X}, in essence under fortified assumptions.

In this connection we note that the lattice <! with 0 e T which in 3.7 and 4.6 occurs in the assumptions does no more occur in the conclusions.

When the assumptions hold true for some such⁽! c (©T©)_L then for the particular choice T = (©T©)_L as well. Thus one has the full theorems when one restricts oneself to⁽! = (6T©)_L. In this case there is also the opposite relation © C (IT'DJL, which in [1] occurs in the overall assumptions. It can thus be said that the added assumption © C (TTI)!.

does not narrow the results of [1].

However, it is an essential point that we removed the overall assump- tions in [1] that © be stable under countable intersections and that f3 be a continuous at 0. On the one hand this produces no loss, because after the fundamentals in MI 6.31 an inner -A- premeasure which for some • == ar is

• continuous at 0 proves to be an inner • premeasure. On the other hand this extension permits to obtain specific results on inner -A- premeasures as in the final part of the last section.

There are some further comments which are related to nontrivial counterexamples. They will be presented in the final section below.

5. Some further comments and counterexamples.

We want to ask two questions. The first question comes from the direct comparison of 3.8 and 4.5. Under identical conditions we have proved

in 3.8: If W C © is nonvoid and totally ordered under inclusion, in 4.5: If 9Jt C ^P(X) is nonvoid and well-ordered under inclusion, then there exists an inner * premeasure (/?: © —^ [0, oo[ which (mini) supports

(3 at 9JI. Of course the question is whether in 4.5 (and hence in 4.6 too) the

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assumption well-ordered can be weakened to totally ordered. We shall see that the answer is no.

Example 5.1. — In a Hausdorff topological space X let 6 = Comp(X) and T = Op(X). Define f3 : G -> [0, oo[ to be {3(0) = 0 and /3(S) = 1 for S ^ 0. After 2.7 and 3.2 we are in the situation of 3.8 and 4.5. Assume that X is infinite. We assert that there exists a nonvoid and totally ordered system TO C ^P(X) of countable subsets of X such that there is no inner -*- premeasure (p : G —^ [0, oo[ which supports (3 at TO.

Proof. — 1) Let (di)i be a sequence of pairwise different points in X, and define / : X -^ [0, oo[ to be /(aQ = 1 - } for / e N and f{x) = 0 for the other x C X. Then TO :== {[/ ^ ^] : 0 < t < 1} C ^(X) consists of nonvoid countable subsets of X, and is nonvoid and totally ordered.

2) Assume that (p : G —> [0, oo[ is an inner ^ premeasure which supports (3 at TO, that is ^ ^ 1 and ^([/ ^ t]) = f3^([f ^ t}) = 1 for 0 < t < 1. Now (p is a Radon premeasure and hence A := (^JBor(X) a finite Borel-Radon measure. It fulfils A([/ ^ t}) = 1 for 0 < t < 1, whereas [/ ^ ^] [ 0 for ^ f 1.

This is a contradiction. D In order to formulate the other question let as usual © and ? be lattices in X with 0 e ©,1 and /3 : © —^ [0, oo[ be isotone with f3(0) == 0.

The results 3.7-3.8 and 4.5-4.6 were under the common conditions that on the one hand 1 C (6T©)J- and 1 separates ©, and on the other hand that /3 is submodular and f3 = /3. Our question is whether these conditions are necessary for the respective conclusions, that is for the existence of inner -*- premeasures with the respective supportive properties.

We restrict ourselves to the conditions on /?, because there were some relevant considerations on G and T in the final part of Section 3. We shall see that the condition that (3 be submodular is indeed a necessary one, whereas the condition f3 = (3 is not, even when one assumes that f3 < oo.

For the latter result we shall exhibit a Hausdorff topological space X and a submodular /3 : G —> [0, oo[ on (5 = Comp(X) such that in case T = Op(X) one has /3 -^ (3 and J3 < oo, while in case T = (©T©)-L D Op(X) one has (3 = /3, which implies that the above conclusions are all valid. We know of no counterexample where ^ = (©T6)-L.

We refer to the related but different considerations in Anger [4] [5], where in the mainstream X is a locally compact Hausdorff topological space. We remark that the condition f3 = f3 becomes a necessary one, and is in fact an immediate consequence of the classical Dini theorem, whenever

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one can, in an appropriate topology on the space of all inner -A- premeasures ip : © —^ [0,oo[, invoke appropriate compact subsets. For these ideas we also refer to the work of Tops0e exemplified in [27] [28] [29].

Remark 5.2. — Assume that for each pair of subsets A c B in G there exists an inner * premeasure (p : © —> [0, oo[ which supports f3 at {A,B}. Then ft is submodular.

Proof.— For fixed A,B e 6 let ^ : G —> [0,oo[ be an inner * premeasure which supports (3 at {AnB, AUB}. Then /3(AuB)+/3(AnB) ==

^(A U B) + (^(A H B) = <^(A) + ^p(B) ^ f3{A) + ft(B). D Example 5.3. — We fix a Hausdorff topological space X which is not discrete and hence not finite, but in which all compact subsets are finite;

thus © = Comp(X) consists of the finite subsets of X. There are well- known countable X of this kind; see for example MI 9.8. Also fix a € X such that {a} is not open. Define (3 : G -^ [0,oo[ to be /3{S) = 0 for S C {a} and l3(S) = 1 for S (JL {a}. Then (3 is isotone and submodular, and A,(T) == 0 for T C {a} and /^(T) = 1 for T (f. {a}. In case ^ = Op(X) therefore /3({a}) = 1 while /3({a}) = 0, so that ft / J3 and /3 ^ 1 < oo. But in case 1 = (©T©)_L = ^(X) one has of course ft = /3. Therefore f3 is as required above.

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Manuscrit recu Ie 1^ septembre 1999.

Heinz KONIG,

Universitat des Saarlandes Mathematik

D-66041 Saarbrucken (Allemagne).

hkoenig@math.uni-sb.de