A comment on unions of rings

A comment on unions of rings

Citation for published version (APA):

Overdijk, D. A., Simons, F. H., & Thiemann, J. G. F. (1978). A comment on unions of rings. (Memorandum COSOR; Vol. 7810). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1978 Document Version:

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of 1-1athematics

PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 78-10 A comment on unions of rings

by

D.A. OVerdijk, F.H. Simons, J'.G.F. Thiemann

Eindhoven, April 1978 The Netherlands

A comment on unions of rings

by

D.A. Overdijk, F.H. Simons, J.G.F. Thiemann

Recently, Broughton and Huff [lJ showed that the union of an increasing se-quence of a-fields cannot be a a-field. It is most remarkable that this fact seems not to have been noted before. However, since the conditions for a class of sUbsets to be a field are weaker than those to be a a-field, the statement that the union of an increasing sequence of fields cannot be a a-field is even more plausible. Unfortunately, the proof of Broughton and Huff makes (only at one place) essentially use of the fact that they consi-der a sequence of a-fields.

In this note we shall give an even simpler proof of the theorem that the union of an increasing sequence of rings cannot be a a-ring. This obviously implies that the union of an increasing sequence of (a-) fields cannot be a a-field.

Throughout, X will be a fixed set and A1 c A2 c . . . will be a strictly in-creasing sequence of rings of sUbsets of X. We put A

=

u:=l An' Then A is a ring, and we have to show that A is not a a-ring.

We start with a modification of the lemma in [lJ.

Lemma. There exists a sequence of disjoint sets (C ) in A such that C ~ A

n n n

for every n.

Proof. We start by taking a set

c

_{1 E A\A 1}, then try to find a set

c

₂ E A\A2 with C

1 n C2

=

0,

etc. If we can perform this construction infinitely many times, we are ready. Therefore, now suppose that after having found

CN' there are no sets in A\A_N+₁ disjoint with C₁

S

=

C

1 u U CN' and define

A* _n

₌

{A E _{A I} A c s},

B* =

{B E

n n

By the construction of S, we have

B*

n for every n > N •

A n

From SEA we conclude that eventually we have SEA ,A

=

{A uBI A E A* n n n' I u

...

_{U CN'} Put B c _(X\S)}

.

the set

- 2

-*

and therefore that eventually the sequence

(A )

is an increasing sequence of

n

fields of subsets of S. But on this sequence the lemma in [lJ can be applied (note that the proof of that lemma also holds

fields), and we obtain a sequence of disjoint

*

F _n ~

A

_n for every n. Then obviously we have F _n

Theorem. A is not a a-ring.

for an increaSing sequence of

co

*

sets (F ) _in u 1

A

with

n n= n

E

A

and F _n ~

A

_n for every n.D

Proof. Suppose

A

is.a a-ring. Let the sequence (C ) be as in the lemma, and

n

{N

1,N2, ..• } be a partition of~ into infinite sets. Put

u

nEN

p

C

n

The sets X are disjoint sets of

AI

and for every p there exists an integer

p

n such that X E A

P P n Since the sequence

(A )

n is increasing, we may suppose

that the sequence

p

(n )

p is increasing as well.

Now for every p we choose a set C c X with

m p

p

00

D

=

u C p=l m p

Again D c

A,

hence D is element of one of the is increasing, there exists a q such that D c the disjointness of the sets.X , we then have

p X q n D

= C

m q E

A

n q m > n , and we put p p

A ,

and since n the sequence (A ) n

A •

Because of n q X c A and q nq

Since n . < m this implies C

q q m E

A

Contradiction.

o

q

Reference.

m

q

[lJ Broughton, A. and B.W. Huff: A comment on unions of sigma-fields. Ann. Math. Monthly 84 (1977), 553-554.