7. The Integral

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?
  • There is a sequence of Riemann integrable functions fn that converges to a function f that is not Riemann integrable.
  • There is a sequence of Riemann integrable functions fn that converges to another Riemann integrable function f but the corresponding sequence of Riemann integrals of fn does not converge to the integral of f.

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 [xj-1, xj]. 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:

  1. 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.
  2. 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
  The outer measure m* is a real-valued set function defined on a space X with the properties:
  1. It is a non-negative set function defined for all subsets of X
  2. m*(O) = 0, i.e. the outer measure of the empty set is zero.
  3. It is monotone increasing, i.e. if A B then m*(A) ≤ m*(B)
  4. It is countably subadditive, i.e. m*( Aj) ≤ m*(Aj)

There are many examples of outer measures. The one that will be particularly useful for us is called Lebesgue Outer Measure, defined as follows:

Definition 7.3.3: Lebesgue Outer Measure
  If A is any subset of R, define the (Lebesgue) outer measure of A as:
m*(A) = inf { l(An) }
where the infimum is taken over all countable collections of open intervals An such that A An and l(An) is the standard length of the interval An.

This is quite a definition. First, you need to find an "open cover" of a set A by intervals An, which simply means that the union of the An cover the set A. That union, by the way, could involve finitely or countably many intervals. Then you need to find the (possibly infinite) sum of the lengths of all these intervals. Finally, you need to find the smallest such sum, taking all possible coverings of A into account. Still confused? I don't blame you, so it's time for some examples:

Examples 7.3.4: Lebesgue Outer Measure
  • 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.

Technically, we have to prove that the Lebesgue outer measure really is an outer measure, i.e. it does have the properties of an outer measure defined above. But the previous example actually proves conditions (1) to (3), while condtion (4) is left as a (technical and tricky) exercise.

We will not really talk about any other outer measure in this chapter, so whenever we say "outer measure" we really mean "Lebesgue outer measure". It turns out by the above examples that the Lebesgue outer measure has the property that m*([a,b]) = m*((a,b)) = b-a, i.e. the Lebesgue outer measure of any interval is its length.

Definition 7.3.5: Measure
  A measure is a non-negative set function defined on a sigma-algebra of sets such that the measure of the empty set is zero and the additional property that it is countably additive, i.e. m( Aj) = m(Aj) for any pairwise disjoint sets Aj.

So, a measure is an outer measure with a domain that no longer consists of all subsets of a space X but is defined on a sigma-algebra of subsets of X, but which is countably additive instead of countably subadditive. The monotonicty property (3) of an outer measure is implied (see example below).

Definition 7.3.6: Sigma Algebra and Measure Space
  A sigma algebra is a collection of subsets of a space X that is closed under complements and countable unions, i.e. the complement of any member of the sigma algebra is again part of the sigma algebra, and the countable union of sets from the sigma algebra is again part of it.
A measure space (X, Σ, m) is a space X together with a sigma algbra Σ of subsets of X and a measure m defined on the sigma-algebra.

There are many measures and outer measures:

Example 7.3.7: Measures
  • Define λ*(E) = 0 if E R is finite and λ*(E) = 1 if E is infinite. Is λ* a measure or an outer measaure?
  • For any A R, define cm(A) = card(A) if A is finite and cm(A) = if A is infinite. Show that that cm is a measure (called counting measure).
  • A measure space (X, Σ, m) with the additional property that m(X) = 1 is called a probability space and the measure m is called probability measure. Suppose we define X = {a, b, c, d}<\math> and Σ = { O, {a, b}, {c,d}, {a,b,c,d} }. Define a probability measure on Σ.

The goal now is to restrict the (Lebesgue) outer measure only to those sets for which it becomes countably additive. This definition is due to Caratheodory. It is relatively straight-forward, but it is not immediately clear as to where his idea comes from.

Definition 7.3.8: Measurable Sets (according to Caratheodory)
  A set E is called measurable if for every set A we have that
m*(A) = m*(A E) + m*(A Ec)
where * is an outer measure. For a measurable set we define the measure of E as m(E) = m*(E).

Note: Since A = (A E) (A Ec) we have by subadditivity m*(A) ≤ m*(A E) + m*(A Ec) Therefore a set E is measurable if m*(A) ≥ m*(A E) + m*(A Ec) since the other inequality comes for free. As it will turn out, the measurable sets are those sets for which the outer measure becomes additive.

Theorem 7.3.9: Properties of Measure
  1. Show that for any two sets A and B we have that m(A - B) = m(A) - m(A B) . What if B A?
  2. The complement of a measurable set is measurable
  3. Every set with outer measure zero is measurable.
  4. The intersecion of two measurable sets is measurable
  5. The union of two measurable sets is measurable
  6. Countable unions and intersecions of measurable sets are measurable
  7. The collection of all measurable sets forms a sigma algebra

Note that the above statements are true for any measure, whereas the next theorem applies to Lebesgue measure in particular.

Definition 7.3.10: Lebesgue Measure
  If we restrict the Lebesgue outer measure to the measurable sets only, the resulting function is called Lebesgue measure.

Note: Another way to define a Lebesgue measurable set would be to define an inner measure m*. Where the outer measure of a set E involves the inf over all open coverings of E, the inner measure of a set E would be the sup over all compact subsets of E. Then we would call those sets measurable for which m*(E) = m*(E). That idea is rather symmetric and fairly straight-forward, but Caratheodory’s approach, while less intuitive, is less abstract and easier to work with, and applies to any outer measure.

We finally prove that moving from Lebesgue outer measure to Lebesgue measure achieves our goal of countable additivity.

Theorem 7.3.11: Properties of Lebesgue Measure
  1. The interval (a, ) is Lebesgue measurable
  2. All intervals are Lebesgue measurable and their measure is their length.
  3. All open and closed sets are Lebesgue measurable
  4. If Aj are Lebesgue measurable, then m( Aj) ≤ m(Aj)
  5. If Aj are Lebesgue measurable and pairwise disjoint, then m( Aj) = m(Aj)

The last of the above properties is what we were after: we wanted the L-measure to be additive and we were willing to pay the price that a measure may not be defined for all subsets. In fact, our measure is defined only for the measurable sets.

After all this theory it is time to consider some easier examples, including the L-measure of some well-known sets that are not intervals (whose L-measure, as we know by now, are their length).

Examples 7.3.12: Properties of Lebesgue Measure
  • Show that the set R and the empty set are L-measurable and find their L-measure.
  • Find the L-measure of a singleton {x} as well as the L-measure of a countable set
  • What is the L-measure of the set Q of all rational numbers and the set I of all irrational numbers inside [0, 1].
  • Find the L-measure of the Cantor middle-third set (if it measurable).
Theorem 7.3.13: Non-Lebesgue measurable set
  1. There are sets that are not Lebesgue measurable.
  2. There are sets for which the Lebesgue outer measure is not additive.

Sometimes we refer to the collection of all open sets, closed sets, and sets that can be derived from them by taking union, intersecions, and complements as part of the Borel sets. But not every Borel set is a union or intersecion of open/closed sets, and unions/intersecions of those, etc., so that the Borel sets are really something new. In fact:

Definition 7.3.14: Borel sets
  The Borel sets are the smallest sigma-algebra that contains the open sets. It includes complements, unions, and intersecions of open sets, and complements, intersecions, and unions of those sets, and so on.

Note: There are unaccountably many Borel sets, but there are not *that* many. In other words, card(Borel sets) = c. Compare this with the cardinality of all Lebesgue measurable sets, which is 2^c (why – see exercises). All Borel sets are (Lebesgue) measurable, but not all Lebesgue measurable sets are Borel sets. So, there are a lot of Lebesgue measurable sets and comparatively few Borel sets.

Before we are done, we need one more result:

Proposition 7.3.15: Monotone Sequences of Measurable Sets
  If { An } is a sequence of measurable sets that is decreasing, i.e. Aj Aj+1 for all j, and m(A1) is finite, then
lim m(Aj) = m(Aj)
If { An } is a sequence of measurable sets that is increasing in the sense that Aj+1 Aj for all j, then
lim m(Aj) = m(Aj)
Examples 7.3.16: 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.

As our final result we show that Lebesgue measurable sets are "almost" open or "almost" closed:

Theorem 7.3.17: Lebesgue Sets are almost open
  If a set A is L-measurable then for any ε > 0 there exists an open set O such that A O and m(O-A) < ε as well as a closed set F such that F A and m(A-F) < ε.

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

  1. 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.
  2. 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
  3. Then we introduced concrete examples of outer measure - the Lebesgue outer measure - and of measure - the Lebesgue measure - and we studied their properties