ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X (1978) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PR ACE MATEMATYCZNE X X (1978)
Z. Lipecki (Wroclaw)
Decomposition theorems îor Boolean rings, with applications to semigroup-valued measures
In several measure-theoretic statements a measure comes into the picture exclusively as a generator of some classes of sets, first of all, th e class of null sets and th a t of non-null ones. Moreover, in a number of cases only purely set-theoretic properties of those classes (like being a cr-ideal and containing no uncountable subclass of pairwise disjoint sets, respectively) tu rn out to be essential in the proofs. The discovery and bringing to light the described phenomenon goes back to Ulam ([13], (B)) and has found continuation in many works (e.g., hTeubrimn [11]
and [1 2] and Cervenanskÿ and Draveckÿ [2]). The recent investigations of set-valued measures (see, e.g., Costé [1]), group-valued measures (e.g., Drewnowski [4] and [5], Herer [7], Musial [9] and [10], and the present author [8]) and various kinds of submeasures (e.g., Dobrakov [3] and Drewnowski [4] and [5]) yield a new motivation for studying set-theoretic nature of other measure-theoretic assertions.
This type of research (*) is done in the present paper. We are mainly concerned with some general decomposition theorems for Boolean rings having origin in measure theory. They are derived in a unified manner, as consequences of two basic assertions (Theorem 1 and Lemma 4). A part from new results (Proposition 1 and Theorem 2) generalizations of some known ones are given (Theorems 3 and 4). In the final section of th e paper we gather measure-theoretic interpretations of the mentioned re
sults.
I. Preliminaries. In this section we give some notation and defini
tions from the theory of Boolean rings to be used throught the paper.
Most of them are more or less standard.
S' stands for a Boolean ring with the operations of join, meet and difference denoted by v, л and \ , respectively. The natural ordering of S is denoted by < and its minimal element by 0.
(*) Supported in part by the Danish Natural Science Research Council.
For E, F cz 8 we put
E a = {a e 8: a лЪ = 0 for any b e E } ,
( E , F} — {a e S: b e F whenever b e E and b < a} (x) . The notation {a}d, where à e 8, is abbreviated to ad.
We say th a t E cz 8 is hereditary if b e E provided b < a for some a e E , i.e., E cz <8 , E y .
Given E cz 8, an element a e 8 \ E is called an E-atom if there are no ax, a2 e 8 \ E with a = axv a 2 and ахл а 2 = 0. The set of all .E-atoms is denoted by A( E) and we p n t
B( E) = {b e 8: a e 8 \ A ( E ) whenever a e 8 and
In the following rrt always denotes an arbitrary cardinal nnmber different from zero. We say th a t E cz 8 is closed under (disjoint) joins of power < m provided for any F cz E (of pairwise disjoint elements) with
|F| < m the join of F exists and belongs to E. (Note th a t then 0 eE. ) A hereditary I cz 8 which is closed under joins of power < m is called an m-ideal. We call 8 a Boolean m-ring if it is closed nnder joins of power
< m. Given E cz 8, we denote by c(E) the 8uslin number of E, i.e., the least m with the property th a t |F| < m for any F cz E of pairwise disjoint elements (2).
In the proofs of Theorem 1 and Lemma I we shall use the following equivalent form of the axiom of choice (cf. [14]):
(D) For any E cz 8 there exists M cz E of pairwise disjoint elements with M an E cz {0} (3).
2. Abstract Lebesgue’s decomposition and atoms. We start with a straightforward generalization of Theorem 1.3 in [11].
Theorem 1 (abstract Lebesgue’s decomposition). I f I cz 8 is closed under disjoint joins of power < rrt, 0 e К c 8 and c ( I \ K ) < m , then there exists m e l with md cz <1, Ky. If, moreover, I \ K Ф 0 and I is hereditary and К is an ideal, then m ф К.
P ro o f. Apply (D) to E A I \ K and denote by m the join of M, which exists in view of the assumptions on I and K . Clearly, m has all the desired properties.
P) In case E and F are the sets of null elements of two different measures on S, a e <E , Fy if and only if on a the second measure is continuous with respect to the first one.
(2) We point out that the last group of terms is often used in a different, essen
tially non-equivalent, sense. The difference consists in replacing “< ” by “< ”. The variant chosen here allows us to gain in generality somewhat.
(3) Nevertheless, under the additional assumption that there exists A: E-> (0, o o )
with X{an)-> 0 for any sequence {an} cz E of pairwise disjoint elements, (D) can he proved more effectively (cf., e.g., [4], Theorem 4.7).
Boolean rings 399 The following lemma is a direct consequence of the definition of an I-atom.
L e m m a 1. I f I <= S is hereditary, then A( I ) c <S \ I , A (I)}.
For a measure-theoretic counterpart of the next result see Corollary 3 in Section 4.
Proposition 1. I f I c S is an m-ideal, 0e Z c f ( 4) and c ( I \ K )
< m , then B( K) <=. B(I).
P ro o f. Since B{K) is hereditary, it suffices to show th a t B( K) r\
r\A(I) = 0 . To this end take a e B ( K ) and put I a — {b e I : b < a}.
Clearly, I a is an m-ideal and e{Ia\ K ) < m a s I a c L Hence, by Theorem 1, there exists, an m e I a with
(*) <= <Ia, -£>•
Suppose a \ m e l . Then a e l , so th a t а фА(1), and we are done. Other
wise a \ m ф! which yields a \ m фК. Hence we have a \ m — axv a 2 for some e S \ K with агл а 2 = 0. In virtue of (*), we get % e $ \ I , so th at а \ т ф А ( 1 ) . Hence, by Lemma 1, а ф A(I).
L e m m a 2. Suppose S is a Boolean m-ring. I f I cz S is an m-ideal, then В (I) is also an m-ideal.
P ro o f. Denote by a the join of an E c S with \E\ < m. Suppose a' < a for some a' e A (I). Then, in view of Lemma 1 and the distribu
tive law ([15], Theorem 1.3), there is b e E with а ' л Ь е А ( 1 ) , so th a t b фВ(1). Hence, if E c= B(I), then a e B ( I ) .
The next assertion is a slight generalization of Theorem I I I.8 in [15].
I t follows immediately from Theorem 1 and Lemma 2.
Proposition 2. Suppose S is a Boolean m-ring. I f I a S is an m-ideal and c ( $ \ I ) < m, then there exists m e B ( I ) with mdr\B{I) c I.
3. Abstract Herer’s and Hahn’s decompositions. Intersection theorem for families of m-ideals (5). We establish two more lemmas, the second of which being, in a sense, a common part of the main results to follow.
L e m m a 3. I f J c S is closed under disjoint joins of power < m, 0 e K cz S and c ( J \ K ) ^ m, then for any a e S \ J there exists a ' e { S \ J ) n ( J , K y with a' < a.
P ro o f. Fix a e S \ J and pu t I = {b e J : b < a}. Choose m satisfying the assertion of Theorem 1. Then a' = a \ m is as desired.
L e m m a 4. I f J y c S, where у e Г, is closed under disjoint joins of power < rrt, 0 e K cz S and c{Jy\ K ) ^ m for у e Г, then there exists c U (S \ J y) n ( J y, K } of pairwise disjoint elements with M d cz Q J y.
УеГ y£r
(4) Note that if 0 $ K , then A ( K ) = 0 , so that B( K) = S.
(°) The material of this section depends only on Theorem 1 from the preceding section.
P ro o f. Applying (D) to E = (J { S \ J y) n ( J y, K) , we get J f с E
у е Г
with М йглЕ = 0 . If a e S \ J Y for some у e Г, then, by Lemma 3, there exists af e E with a1 < a. Hence a $ M d. I t follows th a t M d cz C ] J y.
y e Г
Th e o r e m 2 (abstract H erer’s decomposition; cf. [7]). I f I <= 8 is closed under disjoint joins of power < m and c ( 8 \ I ) < m, then there exists M с ( I , <$, iy y of pairwise disjoint elements with \M\ < m and M d <= /.
P ro o f. Observe th a t c ( I \ 0 , 1}) ^ c ( 8 \ I ) . Hence, by Lemma 4 applied to J Y = I and К = 0 , 1 } , we get M with the desired proper
ties. (That I I f I < m follows from M cz 8 \ I . )
The next result has been proved, for m = by Cervenanskÿ and Hravecky [2].
Th e o r e m 3 (abstract H ahn’s decomposition). I f P, I a 8 and ($ \P )u { 0 ) are closed under disjoint joins of power < m, g[{8\1)\jP j
< m and
(**) a v b e P whenever a e l , Ъ e P and b Ф 0, then there exists p e P n ( S \ P , I > with p d a { 8 \ P ) u{0}.
P ro o f. Applying Lemma 4 to J y = { 8 \ P ) u{0} and К = I, we find Ж cz P n < $ \P , 1} with M d cz ( 8 \ P ) и {0}. Hence, denoting by p the join of M, we get p e P and p d cz ($ \P )u { 0 } . Thus it remains to show th a t p e 0 \ P , I>. To this end fix b e 8 \ P with b < p and denote by bx and b2 the join of {a e M: алЬ e 8 \P } and {a e M: алЬ eP}, respecti
vely. We have b = 6jV&2 and ЪхлЪ2 = 0. Moreover, bx e l and bz e P.
Hence, by (**), b2 = 0 , so th a t b e l .
As the final application of Lemma 4 we establish the following
Th e o r e m 4 (cf. hTeubrunn [1 1], Theorem 3.3, for m = Xx). 8uppose 8 is a Boolean m-ring. I f J Y cz 8, where у e Г, is an m-ideal and c ( 8 \ f ] J y)
у е Г
< m , then there exists Г ' cz Г with [P'| < m and f ] J y — П J y.
у е Г ' у е Г
P ro o f. Let M satisfy the assertion of Lemma 4 for J y and К = C\ J Y.
y e/1 Choose Г' cz Г so th a t M cz [J ( 8 \ J y) n ( J y, К > and |P '| < \M\. We claim
у е Г '
th a t Г ' is as desired. Since M cz 8 \ П J y1 we have \F'\ < m. Moreover,
у е Г
m e ( О J y, П J y}, where m denotes the join of M. Hence, given « e П J y,
p • у е Г ' у е Г у е Г -
We have а л т e p) J y and a \ m e П J y. This shows th a t a e J y.
у е Г ' у е Г ■ у е Г
R e m a rk . For m = the above theorem is formally a very special case of Eeubrunn’s theorem. However, it is not difficult to derive the latter from ours. Indeed, adhering to the notation and terminology of
oo
[11], pu t J i x = {E e£f : П r~n(E) e Ji}. Observe th a t
n = l
Boolean rings 401
(i) M <x££ if and only if S£ <= J{x.
(ii) If J Î satisfies condition (6) in Neubrunn’s theorem, then J l x is a class of null sets (i.e., an Kr ideal).
(iii) If r is ^-non-singular, then J l c= J tx.
In view of (i) and (ii), it follows from Theorem 4 above th a t there exists a countable 9Î <= 9Л with П Л х = П Hence, by (iii), 91 = т Ш1.
л т лет
(Condition (a) in ISTeubrunn’s theorem turns out to be superfluous.)
4. Applications to semigroup-valued measures. We now indicate some of the possible applications of the results of Sections 2 and 3. We concentrate on semigroup-valued measures while the applications to various kinds of submeasures and the like ([3], [4]) are left to the in
terested reader.
In this section 8 denotes a Boolean Xx-ring (i.e., a (7-ring) (6) and G stands for an Abelian Hausdorff topological semigroup (with the iden
tity denoted by 0). By a measure we mean any function /л: 8->G such
oo
th at /4(0) = 0 and /г (a) = ^ y{an) provided {an} a 8 is a sequence of
n = l
pairwise disjoint elements with join a.
Given a measure /л: 8->G, we denote by N (/л) the Nj-ideal °f /«-null elements, i.e., N(ja) = {a e 8: p{b) = 0 for any b < a, b e 8}. We say th a t /л satisfies the countable chain condition (shortly CCC) if с(8\Ж{/л))
< This condition is essentially due to Dubrovskiï ([6], Theorem 3;
see also [12], Theorem II, [5], p. 204, and [9], p. 319).
First we establish two simple consequences of Theorem 1.
Co r o l l a r y 1 (cf. [12], Theorem II). I f /л: S->G satisfies CCC, then there exists m e 8 with /л (a) — /л{алт) for all a e 8.
P ro o f. Applying Theorem 1 to I — 8 and К = N(p), we find m e 8 such th a t /t(6) = 0 provided b e 8 and т л Ь = 0. Hence /л(а) = /4( а л т ) +
~‘r fi (a\ m) = /|( а л л ) for a e 8 .
Co r o l l a r y 2 (Lebesgue’s decomposition; cf. [10], Theorem 9).
Suppose G and H are Abelian Hausdorff topological semigroups and /t : $->6r and v: 8->H are measures. I f v satisfies CCC, then there exist unique measures vx, v2 : 8 ->H with
(***) v = v1 + v2, vx 4. /л and v2J_/b
where, by definition, vx /л and r2_|_/« mean that N (/*) <= N (vx) and md c N (v2) for some m e N( p) , respectively.
(6) This power restriction has become customary although there is no difficulty in considering here (and in many other situations) arbitrary Boolean m-rings and ut-measures for m > sq.
P ro o f. Existence. By Theorem 1, applied to I = N( y) and К = N(v), there exists m e N ( f t ) with mdr\N(y) <= N(v). Hence, putting гх(а) = r ( a \ m) and v2(a) = г( алт) for a e 8, we get measures satisfying (***).
Uniqueness. Let vx and v2 he measures satisfying (***). Then for m , neN(fjt) with md, nd <= N (v2) we have m \ n e Ж(гг) п Е ( г 2), so th a t
(a) m \ n e N(v).
Moreover,
(P) vi(a) = v ( a \ m) and v2(a) — v(a/\m) for a e S .
The assertion now follows from (a) and ((3) and the symmetry of m and n.
A measure y: S-+G is said to be non-atomie provided A [N (y )) = 0 or, equivalently, B(N(y)) = S.
Co r o l l a e y 3 (cf. [1 0 ], Proposition 1 and Example 1). I f measures y x, y 2: 8->G satisfy CCC and are non-atomie, then so is у х~\-y 2-
P ro o f. Consider a measure у : 8-> G xG defined by <p(a) = (ух(а), y 2{a)) for a e 8. Since N (p) = N ( y l) r \ N ( y 2), <p satisfies CCC and is non- atomic. Moreover, N (<p) a N (yx~\- y 2), so th a t the assertion follows by an application of Proposition 1 to К =N(<p) and I = N { y 1-\-y2).
Let us also note th a t Proposition 2 and Theorem 2 immediately imply Theorem 1 of [10] and the Theorem of [7], respectively, while Theorem 4 generalizes Theorem 3 of [6] and Theorem 1 of [9]. We close with another application of Theorem 4.
Co r o l l a r y 4 (cf. [8], Lemma). Suppose G is an Abelian Hausdorff regular topological semigroup. I f a measure y : 8->G satisfies CCC, the?i there exists a sequence {Vn} of closed neighbourhoods of 0 in G such that
O O
N (y) — {a e 8: y(b) e (~) Vn for any b < a, b e 8}.
71 — 1
P ro o f. Denote by P th e family of all sequences {Wn} of closed neigh
bourhoods of 0 in G with T7 n+1 -f Wn+1 c W n. P u t OO
J y = {a e 8: y(b) e О W n for any b < a, b e 8}, where у = {Wn} e Г.
n —1
OO
Since П Wn is a closed subsemigroup of G and y is a measure, J Y is an n~l
Кi-ideal. Moreover, N {у) = П J y by the regularity of G. Hence it follows
у е Г oo
from Theorem 4 th a t there exists a sequence (y j с Г with N(y) = f ] J Yi • P u t Vn = W ^ n W ^ n ... nW%, where {Wln} = y . . Then the sequence {Fn}
is as asserted.
Boolean rings
A d d e d in p ro o f . Belated results, including' a variant of Theorem 2, have been obtained independently by Peter Capek (Mathematica Slovaca, to appear).
R e f e r e n c e s
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INSTITUTE OF MATHEMATICS OF THE POLISH ACADEMY OF SCIENCES WROCLAW BRANCH
