2. Classes of subsets and set function

In order to define the concept of measure, we need to make some preparations. In this section, we will introduce the concepts of semi-algebra, algebra, and -algebra, which are sets closed for some operations.

Definition 2.0.1.

Consider a set and assume that , we say that is a semi-algebra if

such that

Remark 2.0.2. We writhe to be the disjoint union of these sets.

Example 2.0.3. Let and , it is easy to check that is a semi-algebra.

Remark 2.0.4. This example can be extend to , and it inspire us to define the concept of semi-algebra.

Definition 2.0.5. Given a set and we say that is an algebra if

Observation is an algebra is a semi-algebra.

Definition 2.0.6. Given a set and we say that is a -algebra if

Observation is a -algebra is an algebra.
Since algebra is closed to complements and finite intersection, it is obvious that the countable intersection of an algebra is also an algebra, and thus we can discuss the concept of an algebra generated by a set. Similar discussion can be carried out on the -algebra.

Claim 1

Consider a set and a family of algebra over where is an index set. Then is an algebra.

It is clear that since for all . If , then for every , hence for every and therefore . If , then for every hence for every , so .

Claim 2

Given a set and a family of -algebra where is an index set. Then is a -algebra.

It is clear that , and implies that is clear too. The property is similar to that of claim 1.

Definition 2.0.7. Given a set and , we say that an algebra is generated by if

and we denote by .

Let be all algebra over which contain . Then is an algebra generated by .

1. is clear since for all .
2. If is any algebra which contain , then it must be for some and hence .

And is the smallest algebra over which contain . Similarly, we can talk about a -algebra generated by .
The following Lemma shows that the elements of the algebra generated by the semi-algebra can be decomposed into the elements of the semi-algebra .

Lemma 2.0.8.

Consider a set and a semi-algebra and an algebra generated by , then

Proof. () Assume that such that , then since . Hence .

() We construct and claim that is an algebra.

1. since .

2. If , then and where and

3. If , then for . Hence and

since for all . Hence Hence is an algebra. By the definition of , it follows that and it completed the proof.

Then there are two important concepts, the so-called additive function and the -additive function.

Definition 2.0.9. Let and , we define a function and say that is additive if

If and , then

Observation Assume that there exists such that . We can write , then .

Observation Assume that , then

If , then by the additivity of , so

If , then , so

It shows that is an increasing function with respect to the relation "" of sets.
The following example is called a discrete measure, but when we say so, we do not acquiesce in defining the concept of measure. This is like after defining a regular submanifold, we need to prove that it is a manifold; after defining a linear subspace, we need to verify that it is a linear space. Please pay attention to this subtle difference.

Example 2.0.10. (discrete measure)

Given a set and assume that we have a family of points in and a sequence . For , we define where . It is clear that . If such that , then is clear too. Consequently, is additive.

Definition 2.0.11.

Let and , is a function We say that is -additive if

If and with and , then

Observation is -additive is additive. The following example tells us is -additive is additive.

Example 2.0.12. Let and , we define If , it implies that , thus . If , then there are two cases

If for some , then , it implies that since

If for all , then

Thus is additive.

Claim 3

is not -additive.

Take where the decreasing sequence , moreover, we set . but .