Additive Measure Functions

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Concept

A measure function is additive, if the measure of a set is equal to the measure of its finitely many subsets

A measure function is -additive, if the measure of a set is equal to the measure of its possibly infinitely many subsets.

Definition

Additive

The measure function is additive if

2. Let there be a set that is a finite union of disjoint sets with . Then

Sigma-Additive

The measure function is sigma-additive if again

2. Let there be a set that is a union of possibly infinitely many disjoint sets with . Then

Additional Properties

The additive measure of a subset is less or equal to its superset

Examples

The Discrete Measure

Let there be

1. a set:
2. a class of subsets of Omega:
3. a family of points: ,
4. a sequence of non-negative numbers: .

Then we define the discrete measure function as follows:

𝟙

with 𝟙 𝟙

Then without giving the proof, is both additive and sigma-additive.

A non-sigma-additive Measure Function

Let there be

1. a set:
2. a class of subsets of Omega:

Then we define the measure function as follows:

The measure is additive

If the measure was additive, the following would hold:

To proof this, let there be an element with that is a finite union of disjoint sets with .

We can write

Now, there are two cases.

1. Let . Then, we have that . Because there are finitely many , one must contain with . Otherwise, we could always find an but . Therefore

2. Let . Then, we have that . Note that we can write

   Since the intervals are closed on one and open on the other side, (almost) each must occur in exactly two sets , otherwise we could find an that belongs to but not to the finite union. An exception to this rule are and which belong only to a single set, the first and the last respectively. Since all other belong to two sets, they cancel out, we can write

The measure is not sigma-additive

Let us consider the interval which is clearly in . Consider the sequence decreasing to zero: . In other terms . And let , then

We can now write the interval as an infinite union of disjoint sets as follows:

Then by definition of the measure: But since
And , implies that is finite.

In fact, we can write

And this would be equal to

All terms except the last one, in this case, cancel out. Since , the last term, as approaches infinity, goes to zero. Therefore, . If instead x would approach, e.g., 0.1, the result would be and therefore

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Appendix

• Tags:
  • Measure Theory, Measure Functions
• Reference:
  • ; Measure Theory Lecture 2

• Related:

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