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C O L L O Q U I U M M A T H E M A T I C U M

VOL. 81 1999 NO. 1

ON SYSTEMS OF NULL SETS

BY

K. P. S. B H A S K A R A R A O (BANGALORE)

AND

R. M. S H O R T T (MIDDLETOWN, CT)

Abstract. The collection of all sets of measure zero for a finitely additive, group- valued measure is studied and characterised from a combinatorial viewpoint.

Let X be a non-empty set and let A be a class of subsets of X. Then A is a field if X ∈ A and A is closed under the operations of (finite) union and complementation, i.e. A is a Boolean algebra of subsets of X. If A is any class of subsets of X, then a(A) denotes the smallest field containing A. A collection U of subsets of X is a u-system if ∅ ∈ U and U is closed under the operation of proper difference: U

1

\ U

2

∈ U whenever U

1

⊇ U

2

for U

1

, U

2

∈ U. It is easy to show that if U is a u-system such that X ∈ U, and U

1

, U

2

∈ U with U

1

∩ U

2

= ∅, then U

1

∪ U

2

∈ U: a u-system containing X is closed under formation of disjoint unions (and also complements).

Let A

1

, . . . , A

m

and B

1

, . . . , B

n

be finite sequences of not necessarily distinct subsets of a set X. For any k ≥ 1, we define

A(k) = [

A

i1

∩ . . . ∩ A

ik

, B(k) = [

B

i1

∩ . . . ∩ B

ik

,

in each case intending the union of all k-fold intersections: the (i

1

, . . . , i

k

) are k-tuples of distinct indices i

j

. Then we have

A(1) = A

1

∪ . . . ∪ A

m

, A(m) = A

1

∩ . . . ∩ A

m

, B(1) = B

1

∪ . . . ∪ B

n

, B(n) = B

1

∩ . . . ∩ B

n

,

and by convention, we put A(k) = ∅ for k > m and B(k) = ∅ for k > n.

A collection M of subsets of X is an m-system if ∅ ∈ M and whenever A

1

, . . . , A

m

and B

1

, . . . , B

n

are sets in M such that

(∗) A(k + 1) ⊆ B(k) ⊆ A(k) for all k ≥ 1, then

(∗∗)

N

[

k=1

[A(k) \ B(k)] ∈ M, where N ≥ m, n.

1991 Mathematics Subject Classification: 28A05, 28B10.

[1]

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2

K. P. S. B H A S K A R A R A O AND R. M. S H O R T T

Clearly, every field is an m-system, and every m-system is a u-system.

The converse implications do not hold, as is shown in an example given later.

If A is a class of subsets of X, then u(A) and m(A) denote, respectively, the smallest u-system and m-system containing A. Then u(A) ⊆ m(A).

Given a non-empty set X, let Z

X

be the additive group of all functions from X to the integers Z. If A ⊆ X, then the indicator of A is the function 1

A

: X → Z such that 1

A

(x) = 1 if x ∈ A and 1

A

(x) = 0 if x 6∈ A.

Given a collection A of subsets of X, we define S(A) as the subgroup of Z

X

generated by all the indicators 1

A

for A ∈ A.

Lemma 1. If A and B are collections of subsets of X, then S(A ∪ B) = S(A) + S(B).

Lemma 2. Let A be a collection of subsets of X. For any E ⊆ X, we have E ∈ m(A) if and only if 1

E

∈ S(A).

P r o o f. Suppose that 1

E

∈ S(A). Then there are sets A

1

, . . . , A

m

and B

1

, . . . , B

n

in A such that 1

E

= 1

A1

+. . .+1

Am

−1

B1

−. . .−1

Bn

. We see that the sets A

i

and B

j

satisfy condition (∗) in the definition of an m-system, so that E, which is the set in (∗∗), must belong to m(A).

Now let M be the collection of all sets F ⊆ X such that 1

F

∈ S(A). It is easy to verify that M is an m-system containing A, so that u(A) ⊆ M.

The proof gives indication of a useful alternative definition of m-system:

if A

i

and B

j

are sets in M, and 1

E

= 1

A1

+ . . . + 1

Am

− 1

B1

− . . . − 1

Bn

, then E ∈ M.

Lemma 3. Let A be a collection of subsets of X. Then S(m(A)) = S(u(A)) = S(A).

P r o o f. Clearly, S(A)] ⊆ S(u(A)) ⊆ S(m(A)). The inclusion S(m(A)) ⊆ S(A) follows from the preceding lemma.

Example. We show that the concepts of u-system and m-system are in general distinct. Put

Y = {0, 1}

3

, X = {(a

1

, a

2

, a

3

) ∈ Y : a

1

+ a

2

≥ a

3

}, A

i

= {(a

1

, a

2

, a

3

) ∈ X : a

i

= 1} for i = 1, 2, 3.

Then the collection

U = {∅, A

1

, A

2

, A

3

, X \ A

1

, X \ A

2

, X \ A

3

, X}

is a u-system, but m(U) contains the additional set E = {(1, 1, 1), (1, 0, 0), (0, 1, 0)};

we have 1

E

= 1

A1

+ 1

A2

− 1

A3

.

Let A be a field of subsets of a set X and let G be an Abelian group. A

function µ : A → G is a (G-valued) charge if µ(A

1

∪ A

2

) = µ(A

1

) + µ(A

2

)

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SYSTEMS OF NULL SETS

3

whenever A

1

, A

2

are disjoint sets in A. Every G-valued charge µ induces a unique homomorphism ϕ : S(A) → G such that ϕ(1

A

) = µ(A) for every A ∈ A; using the same equation, we see that each homomorphism ϕ : S(A) → G is induced by a charge µ : A → G. Zero sets of group-valued charges are characterised in the

Theorem. Let M be a collection of subsets of a non-empty set X and define A = a(M). The following conditions are equivalent :

(i) there is an Abelian group G and a charge µ : A → G such that M = {A ∈ A : µ(A) = 0};

(ii) M is an m-system.

P r o o f. (i)⇒(ii). Let ϕ : S(A) → G be the homomorphism induced by µ. If A

i

and B

j

are sets in A with µ(A

i

) = µ(B

j

) = 0 and 1

E

= 1

A1

+ . . . + 1

Am

− 1

B1

− . . . − 1

Bn

, then µ(E) = ϕ(1

E

) = 0. The collection M = {A ∈ A : µ(A) = 0} is thus closed under the operation that defines m-systems.

(ii)⇒(i). Define G = S(A)/S(M) and let ϕ : S(A) → G be the standard projection onto the quotient. Define µ : A → G by µ(A) = ϕ(1

A

). By Lemma 2, M = {A ∈ A : µ(A) = 0}.

Quotient groups of the form S(a(A∪B))/[S(A)+S(B)], where A and B are fields, arise naturally in and have been studied for their connection with the problem of joint extensions of group-valued charges (see [1], [2]). With this application in mind, we now prove that the u-system and the m-system generated by the union of two fields coincide.

Theorem. Let A and B be fields of subsets of a set X. For E ⊆ X, we have 1

E

∈ S(A) + S(B) if and only if E ∈ u(A ∪ B). Then u(A ∪ B) = m(A ∪ B).

P r o o f. From Lemma 2 and the inclusion u(A ∪ B) ⊆ m(A ∪ B), we see that 1

E

∈ S(A ∪ B) = S(A) + S(B) whenever E ∈ u(A ∪ B). Now suppose that 1

E

∈ S(A∪B). Then 1

E

= h+k for functions h ∈ S(A) and k ∈ S(B).

Since constant functions in Z

X

belong to S(A) ∩ S(B), it involves no loss of generality to assume that h ≥ 0 and k ≤ 0. Then we have

E =

[

i=0

{x : k(x) ≥ −i} \ {x : h(x) ≤ i},

a finite disjoint union of proper differences of sets of B with sets of A. Thus E ∈ u(A ∪ B).

We have shown that 1

E

∈ S(A∪B) if and only if E ∈ u(A∪B). Lemma 2

then implies that u(A ∪ B) = m(A ∪ B).

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4

K. P. S. B H A S K A R A R A O AND R. M. S H O R T T

REFERENCES

[1] K. P. S. B h a s k a r a R a o and R. M. S h o r t t, Group-valued charges: common ex- tensions and the infinite Chinese remainder property , Proc. Amer. Math. Soc. 113 (1991), 965–972.

[2] R. G ¨ o b e l and R. M. S h o r t t, Algebraic ramifications of the common extension prob- lem for group-valued measures, Fund. Math. 146 (1994), 1–20.

Indian Statistical Institute Bangalore 560059, India

E-mail: kpsbrao@isibang.ernet.in

Wesleyan University Middletown, CT 06457, U.S.A.

E-mail: rshortt@wesleyan.edu

Received 23 May 1994

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