Page 311 - DMTH401_REAL ANALYSIS

Unit 26: Lebesgue Measure

To get the opposite result, we need to show for any  > 0, m*(I) +   b – a. Note that there is a Notes
countable open interval cover {I } of I satisfying
k k
m *(I) +> å (I ).
k
By Heine-Borel Theorem, there is a finite subcover {I } of {I }. Then
n k k
å (I ) > b a (why?)
-
n
k
and it follows that
m*(I) +> å (I )  å (I ) b a.
-
>
k
n
k
Letting   0, m*(I)  b – a. Hence, m*(I) = b – a.
Next, we consider the case where I = (a, b), [a, b), or (a, b] which is bounded but not closed.
Clearly, m*(I)  m* ( I ) = b – a. On the other hand, if  > 0 is sufficiently small then there is a closed
and bounded interval I = [a + , b – ]  I. By monotonicity, m*(I)  m*(I) = b – a – 2. Letting
  0 gives m*(I)  b – a. Hence, m*(I) = b – a.
Finally, if I is unbounded then the result is trivial since in that case I contains interval of
arbitrarily large length.

26.3 Non-measurability

Theorem 3: Let M  () be a translation-invariant -algebra containing all er that intervals, and
m : M  [0, ¥] be a translation-invariant, countably additive measure such

m(I) = (I) for all interval I.
Then there exists a set S  M.
Proof: Define an equivalent relation x ~ y if and only if x – y is rational. Then  is partitioned into
disjoint cosets [x] = {y  : x ~ y}.
By Axiom of Choice and Archimedean property of , there exists S  [0,1] such that the intersection
of S with each coset contains exactly one point.

Enumerate   [–1, 1] into r , r ,.... Then the sets S + r are disjoint and
1 2 i
[0, 1]   (S r )  [–1, 2].
+
i i
If S S  M, then by monotonicity and countable additivity of m we have

1  å m(S r )  3,
+
i
i
which is impossible since m(S + r ) = m(S) for all i  .
i
26.4 Measurable Sets and Lebesgue Measure

As it is mentioned before, the outer measure does not have countable additivity. One may try to
restrict the outer measure m* to a -algebra M  () such that the new measure has all the
properties we wanted.
Definition (Measurability): A set E   is said to be measurable if, for all A  , one has
c
(1) m*(A)= m*(A  E) + m*(A  E ).
Since m* is known to be subadditive, (1) is equivalent to

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