Reference: Introductory Real Analysis by Kolomogorov and Fomin, Dover Pub.
Section 4:System of Sets:
Section 4.1 Rings of sets:
By a system of sets, we mean any set whose elements are themselves sets. Unless the contrary is explicitly stated, the elements of a given system of sets will be assumed to be certain subsets of some fixed set X. System of sets will be denoted by capital script letters like , etc. Our chief interest will be systems of sets which have certain closure properties under some set operations.
A non-empty system of sets is called a ring of sets if and whenever and .
we also have and whenever and . Thus, a ring of sets is a system of sets closed under the operations of taking unions, intersections, differences, and symmetric differences. Clearly, a ring of sets is also closed under the operations of taking finite unions and intersections: and .
A ring of sets must contain the empty set since .
A set E is called the unit of a system of sets if and for every . Clearly, E is unique (why?). Thus, the unit of is just the maximal set of , that is, the set containing all other sets of . A ring of sets with a unit is called an algebra of sets.
Example 1: Given a set A, the system of all subsets of A is an algebra of sets, with unit .
Example 2: The system consisting of the empty set and any nonempty set A is an algebra of sets, with .
Example 3: The system of all finite subsets of a given set A is a ring of sets. This ring is an algebra if and only if A itself is finite.
Example 4: The system of all bounded subsets of the real line is a ring of sets, which does not contain a unit.
The intersection of any set of rings is itself a ring.
Proof 1: This follows from definition 1. QED.
Given any nonempty system of sets , there is a unique ring containing and contained in every ring containing .
If exists, then clearly is unique (why?). To prove the existence of , consider the union of all sets A belonging to and the ring of all subsets of X. Let be the set of all rings of sets contained in and containing . Then, the intersection of all these rings clearly has the desired properties. In fact, obviously contains . Moreover, if is any ring containing , then the intersection is a ring in and hence, , as required. The ring is called the minimal ring generated by the system , and will henceforth be denoted by . QED.
The set containing has been introduced to avoid talking about the “set of all rings containing “. Such concepts as “the set of all sets,” “the set of all rings,” etc. are inherently contradictory and should be avoided. (Recall: Bertrand Russell’s famous set theory paradox).
Semirings of sets:
The following notion is more general than that of a ring of sets and plays an important role in a number of problems (especially in measure theory):
A system of sets is called a semiring (of sets) if
i) contains the empty set ;
ii) whenever and .
iii) If contains the sets A and , then A can be represented as a finite union …..(call this (I)) of pairwise disjoint sets of , with the given set , as its first term.
The representation (I) is called a finite expansion of A, with respect to the sets .
Every ring of sets is a semiring, since if contains A and , then where .
The set of all open intervals , closed intervals and half-open intervals , including the empty interval and the single element sets is a semiring but not a ring.
Suppose the sets where are pairwise disjoint subsets of A, all belong to a semiring . Then, there is a finite expansion
with as its first terms, where , for all .
Proof of lemma 1:
The lemma holds for by the definition of a semiring. Suppose the lemma holds for , and consider sets satisfying the condition of the lemma. By hypothesis,
where the sets are pairwise disjoint subsets of A, all belonging to . Let
By the definition of a semiring, where the sets are pairwise disjoint subsets of , all belonging to . But then it is easy to see that
that is, the lemma is true for . The proof now holds true by mathematical induction. QED.
Given any finite system of sets belonging to a semiring , there is a finite system of pairwise disjoint sets belonging to such that every has a finite expansion where with respect to certain of the sets . (Note: Here denotes some subset of the set depending on the choice of k).
Proof of Lemma 2:
The lemma is trivial for since we only need to set and .
Let the lemma be true for and consider a system of sets in . Let be sets of satisfying the conditions of the lemma with respect to , and let .
Then, by Lemma 1, there is an expansion
where . while, by definition of a semiring, there is an expansion such that
, where .
It is obvious that where k=1,2, …, m.
for some suitable . Moreover, the sets , are pairwise disjoint. Hence, the sets satisfy the conditions of the lemma with respect to . The proof now follows by mathematical induction. QED.
The ring generated by a semiring:
According to Theorem 1, there is a unique minimal ring generated by a system of sets . The actual construction of is quite complicated for arbitrary . However, the construction is completely straightforward if is a semiring, as shown by
If is a semiring, then coincides with the system of all sets A which have finite expansions
with respect to the sets .
Proof of Theorem 3:
First we prove that is a ring. Let A and B be any two sets in . Then, there are expansions
Since is a semiring, the sets also belong to . By Lemma 1, there are expansions as follows:
, where …
Let us call the above two relations as (II).
It follows from (II) that and have the expansions
and hence, belong to . Therefore, is a ring. The fact that is a minimal ring generated by is obvious. QED.
There are many problems (particularly in measure theory) involving unions and intersections not only of a finite number of sets, but also of a countable number of sets. This motivates the following concepts:
ring and algebra:
A ring of sets is called a -ring if it contains the union whenever it contains the sets .
A -ring with a unit E is called a – algebra.
-ring and – algebra:
A ring of sets is called a -ring if it contains the intersection
whenever it contains the sets .
A -ring with a unit E is called a – algebra.
Every -algebra is a -algebra and conversely.
Proof of theorem 4:
These are immediate consequences of the “dual” formulae:
The term Borel algebra or briefly B-algebra is often used to denote a -algebra (equivalently, a -algebra). The simplest example of a B-algebra is the set of all subsets of a given set A.
Given any system of sets , there always exists at least one B-algebra containing . In fact, let
Then, the system of all subsets of X is clearly a B-algebra containing .
If is any Borel-algebra containing and if E is its unit, then every is contained in E and hence,
A borel-algebra is called irreducible (with respect to the system ) if , that is, an irreducible Borel-algebra is a Borel-algebra containing no points that do not belong to one of the sets . In every case, it will be enough to consider only irreducible Borel-algebras.
Theorem 2 has the following analogue for irreducible Borel-algebras:
Given any non empty system of sets , there is a unique irreducible (to be precise, irreducible with respect to ) B-algebra containing and contained in every B-algebra containing .
Proof of theorem 5:
The proof is virtually identical with that of Theorem 2. The B-algebra is called the minimal B-algebra generated by the system or the Borel closure of .
An important role is played in analysis by Borel sets or B-sets. These are the subsets of the real line belonging to the minimal B-algebra generated by the set of all closed intervals .
Exercises to follow,