Časopis pro pěstování matematiky

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1 Časopis pro pěstování ateatiky Beloslav Riečan On the lattice group valued easures Časopis pro pěstování ateatiky, Vol. 101 (1976), No. 4, Persistent URL: Ters of use: Institute of Matheatics AS CR, 1976 Institute of Matheatics of the Acadey of Sciences of the Czech Republic provides access to digitized docuents strictly for personal use. Each copy of any part of this docuent ust contain these Ters of use. This paper has been digitized, optiized for electronic delivery and staped with digital signature within the project DML-CZ: The Czech Digital Matheatics Library

2 Časopis pro pěstování ateatiky, roč. 101 (1976), Praha ON THE LATTICE GROUP VALUED MEASURES BELOSLAV RIECAN, Bratislava (Received July 7, 1975) In the paper we study soe properties of non-negative lattice group valued easures on topological spaces. Naturally enough, this group is assued to satisfy a certain regularity condition. Therefore, the first part is devoted to this condition, a generalization of the Alexandroff theore being proved here. The second part is concerned with the product of easures and the third one with the Kologoroff consistency theore. Let G be an Abelian lattice ordered group, i.e. and Abelian group which is a lattice and which satisfies the iplication: X < J-> X + Z < J T Z. A group valued subeasure fi is a apping /*: 0t -* G, where 0t is a ring of subsets of a space X, nondecreasing, subadditive, fi(0) =» 0 and upper seicontinuous in 0 (i.e. A n \ 0 => => fi(a n ) \ 0). An additive subeasure is called easure. (Of course, every easure is <7-additive.) Definition 1. An Abelian lattice ordered group G is weakly regular*) if it satisfies the following condition: Let a e G, a > 0 and let a n \ 0 (i - oo), then there are such i i9 i that 1=i for no n. As an exaple of a weakly regular group let us take the additive group R of all real nubers. In this case it suffices to choose i k such that 2* *) We say "weakly" since there is a stronger notion of regularity used in [5]. 343

3 Then a й a'/ J=for soe it iplies»... ^ T^ a j-=i2 J which is ipossible. Now let us present two less trivial exaples. Exaple 1. Every linearly ordered group is weakly regular. First we construct the sequence {ij}j*t* Since a\ \ 0 (i -> oo), it is also 2a[ = a\ + a\ \ O, hence there is i t such that 2aJ* < a. Siilarly there is i 2 such that 4a^2 < a and generally ii there is ** such that 2*aJ* < a. If a g J] a]' then -f-i 2 w a j 2"ni l + 2 n a^ n a n n = which is ipossible. -= 2"- 1 2a! B ~ aj? n a» < <2""" 1 a + 2 n " 2 a a = (2 n - 1) a, Exaple 2. Every regular K-space is a weakly regular group. A regular K-space (see [6] Th. VI.5.2) is a linear seiordered space (= Riesz space = K-lineal) which is relatively coplete and such that every sequence of convergent sequences has a coon regulator of convergence. If b n \ 0, then u > O is a regulator of convergence of {b n } nssl iff to any nuber e > 0 there is n 0 such that b n < eu for every n w 0. Hence b n \ O is false iff to any u > O there is e > 0 such that for any n 0 there is n j~ n 0 such that b n < eu is false. Now let a n \ O (i -> oo, n = 1, 2,...) and let u be the coon regulator of convergence of all {a n }% l9 n = 1, 2, Given e > 0 there is! such that If i ^ a n n < M. " 2" wo»0 j=i j-i 2 J then a < eti for every e < 0 which is a contradiction since a > O. In the paper we shall consider only regular easures. Definition 2. Let ^ be a faily of subsets of a set X. We say that ^ is a copact faily if *6 is closed under finite intersections and every decreasing sequence of non-epty sets of < has a non-epty intersection. 344

4 Definition 3. Let & be a ring of subsets of a set X, < c 0t 9 <g a copact faily. Let /i: ^ -* G be a lattice group valued subeasure. We say that ju is inner regular if to any Ee0t there are such sets C n e<& (n = 1, 2,...) that C B c C w+1 c («= 1, 2,...) and li(e - C n ) \ 0. The following theore is a generalization of the Alexandroff theore. Various other generalizations in the real-valued case are found in [4]. Theore 1. Let G be weakly regular, <r-coplete 9 *) Abelian lattice-ordered group. Let X be a topological space, 0t a ring of subsets of X. Let fx : 0t -» G, /i(0) = O be onotone 9 subadditive and inner regular. Then p, is upper seicontinuous in 0. Proof. Let A n \ 0 (i.e. A n z> A n+i (n = 1, 2,...) and f] A n = 0). We want to «==-I prove that fi(a n ) \ O. Let us prove it indirectly. Since G is <r-coplete, there is a > 0 such that fi(a n ) J> a. Since \i is inner regular, to any n there are C n e # (i = 1, 2,...) such that C'czC^cz^, 0=1,2,...) and li(a n -C n )\0 (i -+ oo). Put <* = M^«) > <*» = K^n ~ Cn) and choose it, i 2,... by Definition, 1. Now put D t = C[\ D 2 = C< 2 n Cj 1,..., A, = C B» n C^1 n... n Q,... Then D B ^, D n z> D n + t (n = l,2,...). We prove that > * 0 («= 1,2,...): If D n = 0, then which is ipossible. a S n(a n ) fi((\j(aj - Cj9)u(nC}9) =^ i=i 1=1 = f ^ - <#) + a(d n ) = f/<a y - Cy) = ta'/ Since D n z> D.+t,!) e 9 D n # 0 (n = 1. 2,...), we have f] D n = 9. But!> c A h»=i (n = 1, 2,...), hence also f) A n # 0, which is a contradiction. oo *) I.e. every bounded countable set has the supreu. 345

5 Now we want to prove a theore oh the product of two easures. Usually the product of two easures \i, v is defined as such a easure X in the cartesian product that X(E x F) = \i(e)v(f) for all E, F fro the corresponding doains. However, in our general group G we need not have any product. Hence we shall assue that there are given three groups G t, G 2, G and a apping n : Gi x G 2 -> G satisfying soe conditions. We shall need the following three siple conditions: 1. n(a + b, c) = n(a, c) + n(b, c), n(a', V + c') = n(a!, V) + n(a',c') for all a, b, a' eg t, c, V, c' eg If a ^ O, b = O, a e G x, b e G 2, then n(a, b) ^ If a n \ 0, b n \ 0, a n e G L,fe e G 2 (n = 1, 2,...) then n(a n, b n ) \ O. Theore 2. Let 0t t or 0t 2 be rings of subsets of X t or X 2 respectively. Let \i\0t t -+ G t, v \0t 2 ~* G 2 be inner regular easures. Let G be weakly regular, a-coplete, Abelian, lattice-ordered group. Then there is exactly one G-valued easure X defined on the ring 01 generated by the faily Q> = [E x F; Ee0t x, F E 0t 2 } and such that for all Ee0t t, F 0t 2. X(E x F) =* n(ii(e), v(f)) Proof. Define first X 0 \ 9 -» G by the forula X 0 (E x F) = n(n(e), v(f)). Evidently X 0 is additive, onotone, X o (0) = 0. Hence we can extend X 0 to a function A : 0t -* G by the forula A(U^) = ZA 0 (^) i=i. t-i. where ALj are disjoint sets fro & (i = 1,..., n). The function X is also additive, nonnegative (and therefore onotone and subadditive). It suffices to prove that X is upper seicontinuous in 0. Let ^j, * 2 be copact failies of subsets of X t or X 2 respectively. Let <% consist of allfiniteunions of sets of the for C x D where Ce<$ u De<$ 2. Then < is a copact faily. 346 Now let ie Then A = U -4* = U ( i i*i r*-r ' x -^i)* where ^4* are pairwise disjoint.

6 Since E t e dt t and fi is inner regular, there are C* e C t (n = 1, 2,...) such that and C n tczc1 +l cei (n = 1,2,...) fi(ei- C?) \ 0 (n-oo). Siilarly there are D n ie c 1 (n = 1,2,...) such that and D?c DJ +1 c F f (n = 1,2,...) v(f* - D?) \ 0 (n - oo). By the third property of n we have X(A t -C1 x >?) = A(( * x F f ) - (C? x D?)) g 4l4?i - C 0> KF,.)) + n(ix(e^ v(f t - Dj)) s, O (n -* oo). Put K,, = U (CT x D1) (n = 1, 2,...). Then # e«f, K B = K n+1 ca(n and i=l A(A - K ) = A(lMi - U(e? x ->?)) = i=l i=l = A( U (A, - CI x DJ)) - S^i ~ C? x DJ) s, 0 i=l i=l = l,2,...) if n -^ oo. Hence A is regular and the proof is coplete. Reark. A special case of Theore 2 is Theore 2 in [3]. Let {X t } tet be a faily of topological spaces. Denote by F the set of all finite subsets of T. For any a e F put X a = X X t. If a, j8 e F, a z> /? then 7c a^ denotes the tea projection n afi : K a -» X fi. Every X a is a topological space with the product topology and every n afi is a continuous apping. Let G be -a weakly regular Abelian /-group. Now we shall assue that we are given a consistent faily of inner regular G-valued easures {^} aer. Of course, regularity is taken with respect to the copact faily of copact subsets of the corresponding space. Hence for every <x ^> f} and every E e 0t p (dtp is the doain of ftp) we have *;,'( ) e a, n a (n; fi l (E)) = fip(e). 347

7 In this case the projective liit of the projective syste (X a, 0t a, p a, n afi ) is (X,0t,p,n a ), where X -\X t, n a is the projection n a :X -+ X a, 9t = {n l a (E); tet Ee0t a,ab r}, p(a) = p(n % a (E)) = p a (E). It is not difficult to prove that the definition of p is correct (p does not depend on the choice of a), 0t is a ring and that p is additive, onotone, p(9) = 0. The only proble is whether p is a-additive, i.e. whether (X,, p, n a ) is the projective liit of the syste in the category of easure spaces (see [1], [2]). Theore 3. Let G be a weakly regular, a-coplete, Abelian l-group. The function p defined above is a easure and (X,$,p,n a ) is the projective liit in the category of easure spaces. Proof. To prove that p is cr-additive it suffices to prove that p is upper seicontinuous in 0. Let c a denote the faily of all copact sets in X a. Put <^ = {7i; 1 ( ); et«er}. Evidently p is inner regular with respect to (. We prove that ^ is a copact faily. Let C n e<$, C DC B+1, C n *0 (n = l,2,...). Then C n = ^(.D,), D n e<$ an (n = 1,2,...). The set U <x n is countable. Put»=i Consider the sequence IK = {h>t 2,t 3,...}. 11=1 {n M {C u )}? l. If t t 4 <x n, then n {tl) (C n ) = X tl. If t t e a n, then {t t } cz a, hence and this is a copact subset of X ri - Moreover, the sequence {n {tl) (C n )} n^i is decreasing, therefore oo H n [ti) (C ) * 0. -=l Denote by x an eleent of f\ n {ti )(C n ) and repeat the procedure with the second coordinate t 2 : n=i E n - 7t{r 2 )(C B n *(7,>({*?,})) Then E => +., is closed (n = -> 2». ) and is copact if f 2 e a. Hence E, * 0. "i

8 Denote by x 2 an eleent of f) E n. Repeating this procedure we obtain a sequence»-~i such that x x x H* *2' * * *V <en* { jc.n'na-s})) * -1,2,..., 11=1 i=l hence to any n there is x e C n such that Define x by the following forula: Now we assert that x ef\ C n.»=i x fl = x ri, Xf 2 = x f2,..., x tk = x flc. (x ), = x?, if te(j<x n (x ) t = an arbitrary eleent of X t, if f \Joc n. Take arbitrary n and k such that a c {f 1$..., f*}. We know that there is x e C such that x fl = x ri, x f2 = x r2,..., x tk = x fk. Put a n = {f it,..., t j }. Since x e C n9 n^jx) e D n, hence (x^,..., x?j = (x fii,..., x tj J ed n. But it follows that n an (x ) e D n9 i.e. x e n^l(d H ) = C. We have proved that ^ is a copact syste. By Theore 1 n is upper continuous in 0, i.e. \x is a easure. References [1] Dinculeaж N.: Projective Hits of easure spaces, Revue Rouaine 14 (1969), [2] Métivier M.: Liites projectives de esures. Martingales. Applications. Annali di Mat. pura ed applicata 63 (1963), [3] Riečan B.: On the product of vectoг easures with values in seioгdered spaces, Mat. časopis 21 (1971), [4] Rieěanová Z.: A note on a theore of A. D. Alexandroff, Mat. őasopis 21 (1971), [5] VolaufP.: Extension of group-valued easures, to appear. [6] Vulich B. Z.: An introduction to the theory of seiordered spaces (Russian), Moscow Authoґs address: Bratislava, Mlynská dolina (Prírodovedecká fakulta UK). 349