Given a nonempty abstract set

Let

We also
consider for a finite, finitely subadditive outer measure

Lastly,
we consider the general problem of constructing a finite, finitely subadditive
outer measure from an arbitrary family,

We introduce the necessary outer measure, and covering class definitions, and note some known properties that we shall need.

The
definitions and notations are standard and are consistent with those found in,
for example, [

Throughout this section and the rest of the paper,

A covering class

A covering class

Let

It is known that if a covering class coallocates itself, then it is a normal covering class.

Let

It is always true that

It is well known that

An outer measure

Let

If for all

If

Let

In
[

We
begin by recalling the basic construction and properties of

(The functions

We now
determine conditions for

It is known
that if

If

Let

Suppose

Let

Combining
Theorems

If

We recall that given an outer measure

Let

It is known
that when a finite, finitely subadditive outer measure

Let

We first
show that the hypothesis that each

Let

By (

We next show that
when

Let

An important
question to consider is whether or not

If

It suffices
to show by a theorem in [

Let

Now

Therefore, by (

Consequently,

In
this section, we again consider a covering class

Let

Let

Now suppose

Next, consider any

To show that

Combining inequalities (

Since

We now
determine conditions when the inequality of Theorem

If

We show that

Now

Since

Since

If

The
hypothesis that

Since

We next show

We also have

We next show

But

Thus far, in
order to have the equality

We have the following theorem.

Suppose
each

The result follows from [

The results
of this section prove useful in studying the restriction of a finite, finitely
subadditive outer measure to the algebra generated by the covering class, when
the sets of the covering class are measurable, which will be investigated in a
subsequent
paper. Although Theorem

In this
section, we consider a nonempty set

For
emphasis, let

It follows by a standard argument that

We observe that if

The
following theorem is known, and its proof can be found in [

If

Our
motivation for what follows comes from the characterization of the measurable
sets for a

Let

Let

Since

The hypothesis on

Thus we see that a sufficient condition for a set

An example may help to illustrate the theorem.

Let

We return to
characterizing those sets

Let

Let

Let

Now

Suppose

This observation establishes the following theorem.

If

We see
that our result is analogous to what happens in the case of constructing an
outer measure from a measure in the general case, see for instance Munroe [

In
summary, to obtain a set

If

We note
that if

The author wishes to thank the referee for several useful suggestions leading to the improved clarity of the paper, and to the editor for his courtesy. The questions considered for this paper were presented to the author by the late Dr. George Bachman, the author’s teacher and friend.