7.3. Measures
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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 |
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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?
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There is a sequence of Riemann integrable functions fn
that converges to a function f that is not Riemann integrable.
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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.
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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:
- 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 |
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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.
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Note that it is possible to define other outer measures without using the
concept of length of intervals, so the above outer measures is but one of many.
In fact, this particular outer measure is called the Lebesque outer
measure, but we will not talk about any other outer measure in this chapter, so
whenever we say "outer measure" we really mean "Lebesque outer measure".
| Examples 7.3.3: Outer Measure of Intervals |
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-
Find the outer measure of the empty set O, and prove that
m*(A)
m*(B)
for all
A B.
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Find the outer measure of a closed interval [a, b]
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Find the outer measure of an open interval (a, b)
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Find the outer measure of an infinite interval
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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.
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Outer measure is defined for all sets in
R and has some of
the properties we wanted our concept of measure to have.
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 |
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Show that the outer measure of a single point is 0, and the outer measure
of a countable set is also 0.
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The set [0, 1] is not countable.
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Outer measure is subadditive, but not additive.
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Define a function f on the set X = {a, b, c}
by setting f(O) = 0, f(X) = 2,
and f(A) = 1 for any other subset A of X.
Could this function be called an outer measure, i.e. is it a non-negative set
function whose domain is P(R) and which is subadditive? Is it
additive?
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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.
Actually, 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. Another idea is to define a set
E as measurable if it is "almost" an open set, in the sense that for every
> 0
there exists an open set
U R
such that
m*(U \ E) .
This is an idea known as one of Littlewood's three principles, which is
helpful enough, but we will choose yet another approach due to Caratheodory that
is (hopefully) less abstract and (somewhat) easier to work with.
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 |
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Show that the empty set, the set R, and the complement of a
measurable set are all measurable.
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Show that every set with outer measure 0 is Lebesgue measurable.
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Show that the union of two measurable sets is measurable.
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Show that the intersection of two measurable sets is measurable.
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Show that the interval (a,
 )
is measurable.
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The next result shows that a measure has all of the properties of a length: the
measure of an interval is its length, and the measure of countably many disjoint
sets it is additive. But, as we mentioned before, the price we have to pay
for that is that it is no longer defined for all sets.
Sometimes we refer to the collection of all open sets, closed sets, and sets
that can be derived from them as Borel sets. Its significance for our
discussion is that all Borel sets are measurable. So, instead of listing open sets,
closed sets, unions of them, intersections of them, etc. we refer to them collectively
as Borel sets. However, it is important to note that not every Borel set is
a union or intersection of open/closed sets, and unions/intersections of those,
etc., so that the Borel sets are really something new. But these details would
lead us to far afield.
| Theorem 7.3.9: Borel sets are measurable |
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The collection of Borel sets is the smallest sigma-algebra which contains all of
the open sets. Every Borel set, in particular every open and closed set, is
measurable.
Proof
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After all this theory it is time to consider some easier examples, including
the measure of some well-known sets that are not intervals (whose measure, as
we know by now, are their length).
| Examples 7.3.10: Properties of Measure |
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Show that for any two sets A and B we have that
m(A - B) = m(A) -
m(A B)
.
What if B A?
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What is the measure of the set Q of all rational numbers and the
set I of all irrational numbers inside [0, 1].
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Find the measure of the Cantor middle-third set (if it measurable).
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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, as we will see shortly. First, let's take
a quick look into sets that arise in conjunction with functions? First, let's
establish some equivalency conditions.
Note that this does not say that each or any of these sets are indeed
measurable. It does say that if any of these four sets are measurable,
then the remaining ones are also measurable. An easy case in which one of these
sets is clearly measurable is described next.
So here we have further examples of sets that are measurable. But not
all is roses, and you can't always have your cake and eat it, too: we do have to
pay for the property that measure is (countably) additive with the fact that
not every set is measurable.
It is therefore true that not every set is measurable, but it is fair
to say that most sets are.
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:
| Examples 7.3.15: Monotone sequences of measurable sets |
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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.
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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.
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