7.3. Measures

The Riemann integral is certainly useful (and complicated) enough, but it does have a few limitations and oddities:

Examples 7.3.1:Oddities of Riemann Integral

What happens when you change the value of a Riemann integrable function at a single point?
Is it true that a function that is constant except at countably many points is Riemann integrable?
What is the difference between Riemann integrable functions and bounded continuous functions?
Can you take a Riemann integral over anything else but an interval?
Could you define a Riemann integral of a function whose domain is not R?

We therefore want to define another concept of integration that is more general than the Riemann integral, yet retains the "good" properties of that integral.

One of the limitations of the Riemann integral is that it is based on the concept of an "interval", or rather on the length of subintervals [x_j-1, x_j]. We therefore need to find a generalization of the "length" concept of a set in the real line. That new "length" concept, which we will call "measure", should satisfy two key conditions:

The new "measure" concept should be applicable to intervals, unions of intervals, and to more general sets (such as a Cantor set). Ideally, it should be defined for all sets.
The new "measure" concept should share as many properties as possible with the standard length of an interval, such as:

the 'measure' of a set should be non-negative
the 'measure' of an interval should be the length of that interval
the 'measure' of countably many disjoint sets should be the sum of the 'measures' of the individual sets

To define what will eventually be called Lebesgue Measure, we follow a two-stage strategy:

Stage One:: We will define a concept extending length that is defined for all sets (to satisfy condition 1 above)
Stage Two:: We will modify that concept so that it looks as close as possible to the standard length concept (to satisfy condition 2 above)

The stage-one concept is called outer measure, defined as follows:

Definition 7.3.2: Outer Measure

If A is any subset of R, define the (Lebesgue) outer measure of A as:
m^*(A) = inf { l(A_n) }
where the infimum is taken over all countable collections of open intervals A_n such that A A_n and l(A_n) is the standard length of the interval A_n.

Examples 7.3.3: Outer Measure of Intervals

Find the outer measure of the empty set O, and prove that m^*(A) m^*(B) for all A B.
Find the outer measure of a closed interval [a, b]
Find the outer measure of an open interval (a, b)
Find the outer measure of an infinite interval
Find the outer measure of the set A of all rational numbers in [0, 1]. Also show that for any finite collection of intervals covering A we have that the sum of their lengths is greater or equal to 1.

Outer measure is defined for all sets in R and has some of the properties we wanted our concept of measure to have.

Proposition 7.3.4: Properties of Outer Measure

Outer measure has the following properties:

Outer measure m^* is a non-negative set function whose domain is P(R), i.e. the power set of R.
The outer measure of an interval is its length.
Outer measure is countably subadditive, i.e. if { A_n } is a countable collection of sets, then
m^*( A_n) m^*(A_n)

Proof

Since outer measure is defined for all sets in R, our first key condition above is satisfied. But outer measure is not quite similar to a length, because it is only subadditive ( m^*(A B) m^*(A) + m^*(B) ), not additive ( m^*(A B) = m^*(A) + m^*(B) for disjoint sets A and B). Therefore one of the requirements of the second key condition is not satisfied.

Examples 7.3.5: Properties of Outer Measure

Show that the outer measure of a single point is 0, and the outer measure of a countable set is also 0.
The set [0, 1] is not countable.
Outer measure is subadditive, but not additive.

To make our outer measure additive, something has to give. We decide to restrict the domain to gain countable additivity, which will result in our stage-two definition.

There are several ways to restrict an outer measure. A somewhat intuitive way, perhaps, is to define an inner measure m_* that is similar to the outer measure but involves a sup over suitable sets. Then sets for which m^*(A) = m_*(A) are called measurable and the restriction of the outer (or inner) measure to the measurable sets is called a measure.

Using an outer and inner measure to define a measure does not have many advantages, and as a definition it is difficult to deal with. We prefer another approach that is due to Caratheodory to define measurable sets.

Definition 7.3.6: Measurable Sets and Lebesgue Measure

A set E is (Lebesque) measurable if for every set A we have that
m^*(A) = m^*(A E) + m^*(A comp(E))
If E is measurable, the non-negative number m(E) = m^*(E) is the (Lebesgue) measure of the set E.

Since outer measure is subadditive, we have

m^*(A) = m^*( (A E) (A comp(E)) )
m^*(A E) + m^*(A comp(E))

Therefore, to show that a set E is measurable it is sufficient to prove that

m^*(A) m^*(A E) + m^*(A comp(E))

The motivation for this definition is that it ensures that for two disjoint measurable sets E and F we have that

m(E F) = m(E) + m(F)

as we will show later, i.e. measure is additive.

Examples 7.3.7: Measurable Sets

Show that the empty set, the set R, and the complement of a measurable set are all measurable.
Show that every set with outer measure 0 is Lebesgue measurable.
Show that the union of two measurable sets is measurable.
Show that the intersection of two measurable sets is measurable.
Show that the interval (a, ) is measurable.

The next two results shows that a measure has all of the properties of a length, but is no longer defined for all sets.

Theorem 7.3.8: Properties of Lebesgue measure

All intervals are measurable and the measure of an interval is its length
All open and closed sets are measurable
The union and intersection of a finite or countable number of measurable sets is again measurable
If A is measurable and A is the union of countable number of measurable sets A_n, then m(A) m(A_n)
If A is measurable and A is the union of countable number of disjoint measurable sets A_n, then m(A) = m(A_n)

Proof

Examples 7.3.9: Properties of Measure

Show that for any two sets A and B we have that
m(A - B) = m(A) - m(A B) . What if B A?
What is the measure of the set Q of all rational numbers and the set I of all irrational numbers inside [0, 1].
Find the measure of the Cantor middle-third set (if it measurable).

Outer measure is defined for all sets, and according to the above list of properties, most "common" sets such as intervals, closed or open sets, unions and intersections of measurable sets are all measurable. But we have to pay for the property that measure is (countably) additive with the fact that not every set is measurable.

Proposition 7.3.10: Not all Sets are Measurable

There are sets that are not (Lebesgue) measurable, i.e. not every set is (Lebesgue) measurable.
Proof

To summarize, we introduced a new concept called measure in two stages, each of which had a typical "good news, bad news" property.

In the first stage we defined outer measure.
- good news: outer measure is defined for all sets.
- bad news: outer measure is not additive, i.e. it is not quite comparable to a length.

In the second stage we defined measure by restricting outer measure to the measurable sets.

good news: measure is additive, i.e. it is a good generalization of length.
bad news: measure is not defined for all sets

Before concluding this section, we want to prove one more result that will be important for the next section:

Proposition 7.3.11: Monotone Sequences of Measurable Sets

If { A_n } is a sequence of measurable sets that is decreasing, i.e. A_j A_j+1 for all j, and m(A₁) is finite, then
lim m(A_j) = m(A_j)
If { A_n } is a sequence of measurable sets that is increasing in the sense that A_j+1 A_j for all j, then
lim m(A_j) = m(A_j)

Proof

Examples 7.3.11: Monotone sequences of measurable sets

For decreasing sets we had to assume that m(A₁) was finite. Show that without this assumption the statement in the previous proposition is false.
Find the measure of the Cantor middle-fifth set, i.e. the set obtained by using a Cantor-set construction but removing the middle-fifth instead of the middle-third at each stage.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 26, 2007

*Definition 7.3.2: Outer Measure*
	If A is any subset of R, define the (Lebesgue) outer measure of A as: m^(A) = inf { l(A_n) }* where the infimum is taken over all countable collections of open intervals A_n such that A A_n and l(A_n) is the standard length of the interval A_n.

*Proposition 7.3.4: Properties of Outer Measure*
	Outer measure has the following properties: Outer measure m^* is a non-negative set function whose domain is P(R), i.e. the power set of R. The outer measure of an interval is its length. Outer measure is countably subadditive, i.e. if { A_n } is a countable collection of sets, then m^( A_n) m^(A_n) Proof

*Examples 7.3.5: Properties of Outer Measure*
	Show that the outer measure of a single point is 0, and the outer measure of a countable set is also 0. The set [0, 1] is not countable. Outer measure is subadditive, but not additive.

*Definition 7.3.6: Measurable Sets and Lebesgue Measure*
	A set E is (Lebesque) measurable if for every set A we have that m^(A) = m^(A E) + m^(A comp(E))* If E is measurable, the non-negative number m(E) = m^(E)* is the (Lebesgue) measure of the set E.

*Examples 7.3.7: Measurable Sets*
	Show that the empty set, the set R, and the complement of a measurable set are all measurable. Show that every set with outer measure 0 is Lebesgue measurable. Show that the union of two measurable sets is measurable. Show that the intersection of two measurable sets is measurable. Show that the interval (a, ) is measurable.

*Examples 7.3.9: Properties of Measure*
	Show that for any two sets A and B we have that m(A - B) = m(A) - m(A B) . What if B A? What is the measure of the set Q of all rational numbers and the set I of all irrational numbers inside [0, 1]. Find the measure of the Cantor middle-third set (if it measurable).

*Proposition 7.3.10: Not all Sets are Measurable*
	There are sets that are not (Lebesgue) measurable, i.e. not every set is (Lebesgue) measurable. Proof

*Examples 7.3.11: Monotone sequences of measurable sets*
	For decreasing sets we had to assume that m(A₁) was finite. Show that without this assumption the statement in the previous proposition is false. Find the measure of the Cantor middle-fifth set, i.e. the set obtained by using a Cantor-set construction but removing the middle-fifth instead of the middle-third at each stage.