Measurable Sets and Functions

The first question, naturally, is: why one would be interested in another type of integral ? The Riemann Integral certainly seems complicated enough, being defined as a limit of Upper and Lower sums. While a more thorough comparison between the Riemann and the Lebesgue integral will be given in a later section, here are a few things to ponder:

• The only functions that are Riemann integrable are the continuous functions, or those functions that have not too many points of discontinuity. Therefore, there seems to be not much difference between Riemann integrable and continuous functions, which might render one of the two concepts as somewhat superfluous.
• Riemann integrals can only be taken over intervals, or union of intervals. You can not integrate a function over, say, the Cantor middle third set. Since we know that many closed sets can be as complicated as the Cantor set, we might try to define a concept of the integral that works for those sets as well.
• Riemann integrals can be defined for functions from R to R, and they depend on the basic (and complicated) structure of the real line. On the other hand, it is easy to define functions that have other spaces as their domain (sequences, for example, are functions from N to R). The Riemann integral is very difficult to extend to other spaces, and it would be nice to have a concept of integration similar to the Riemann integral that works also, and just as easy, on more abstract spaces.

When trying to establish a more general concept of 'integral' we first need to find a generalization of 'length' of a set in the real line. Since the only sets that have an obvious length in R are intervals, or unions of intervals, our new concept should satisfy two key conditions:

• the new 'length' concept should be applicable to intervals, unions of intervals, and in addition to more general sets
• the new 'length' concept, when applied to intervals, should yield the familiar concept of the length of that interval

This new concept of 'length' will be the measure of a set. To define it, we have to use several steps:

Definition: Basic and Elementary sets in R

• A basic set in R is a set that is in one of the following forms:
  • (a, b), [a, b] (open and closed intervals)
  • [a, b), (a, b] (half-open intervals
• An elementary set in R consists of finite unions of basic sets.

On these elementary sets we can define the concept of length, or measure:

Definition: Measure of Elementary Sets

• If A is a basic set, we define the measure of the set A as (A) = b - a where a and b are the endpoints of the basic set A. The measure of the empty set is defined to be zero.
• If A is an elementary set, then A = , where each set is a basic set. Define the measure of the set A as (A) = ( ) where ( ) = - , and , are the endpoints of the basic set .

Examples:

• Two examples here

Now that we have the concept of measure, we need to determine some of its properties:

Proposition: Properties of Measure and Elementary Sets

• The union and intersection of two elementary sets is again an elementary set.
• The measure defined on all elementary sets A has the properties:
  • (A) is real and non- negative
  • (A) is additive, i.e. if A, B are two disjoint elementary sets then (A B) = (A) + (B).
  • If A is an elementary set and {}is a collection of countably many elementary sets such that A , then (A) ()

Examples:

• Several examples here

At this stage we have defined a concept of length, called measure, that applies to the basic and elementary sets, and agrees with the concept of length on those sets. In other words, its nothing more than a new word for an old concept. The next step is to extend this definition to more general sets, and thereby really obtaining something new.

Definition: Inner and Outer Measure

• If A is any subset of R, define the outer measure of A as:
  • (A) = inf{ ( ): A }
  • where the infimum is taken over all coverings of A by a finite or countable system of basic sets
• If A is any subset of R, define the inner measure of A as
  • (A) = sup{ ( ): A }
  • where the supremum is taken over all coverings of A by a finite or countable system of basic sets

Examples:

• Several examples here

Finally, we are ready to define our new concept of measure for general sets:

Definition: Measurable sets

• A set A is said to be Lebesgue measurable if
  • (A) = (A)
• If A is measurable, the non-negative number m(A) = (A) is the Lebesgue measure of the set A.

Examples:

• Two examples or so here

However, we now have the problem that we have two definitions of measures: one definition of (A) for elementary sets A, and another definitions of m(A) for a measurable set A. In fact, we don't even know whether there are any 'measurable sets'. Our next result will reconcile these definitions and show that this new measure has all of the properties of our previous simple measure.

Theorem: Properties of Lebesgue measure

• All elementary sets are measurable
• If A is an elementary set, then (A) = m(A), i.e. Lebesgue measure and the usual concept of length agree when both are applicable.
• 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 , then m(A) m().
• If A is measurable, and A is the union of countable number of disjoint measurable sets , then m(A) = m().

Examples:

• Several examples here

Now we have defined a new concept of length that does satisfy the two key properties introduced at the beginning of this section:

• it applies to the basic sets and to a more general class of sets called the measurable sets
• when applied to the basic sets it has the same property as the familiar concept of length

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

Proposition: Monotone Sequences of Measurable Sets

• If {} is a sequence of measurable sets that is decreasing in the sense that for all j, then lim m() = m(A), where A = ( )
• If {} is a sequence of measurable sets that is increasing in the sense that for all j, then lim m() = m(A), where A = ().

Examples:

• Three examples here