Page 315 - DMTH401_REAL ANALYSIS

Unit 26: Lebesgue Measure

Theorem 6: Littlewood’s 1st Principle Notes
Every measurable set of finite measure is nearly a finite union of disjoint open intervals, in the
sense

 If E is measurable and m(E) < ¥, then for any  > 0 there is a finite union U of open intervals
such that m*(U  E) < . (Clearly, the intervals can be chosen to be disjoint.)
 If for any > 0 there is a finite union U of open intervals such that m*(U  E) < , then E is
measurable. (The finiteness assumption m*(E) < ¥ is not essential.)

Proof: If we can prove (1), (2), and (4) are equivalent, then it is easy to see that (2) and (3) are
c
equivalent, because one implies another by replacing E with E . Similarly, (4) and (5) are
equivalent.
To show (1)  (2)
We first consider a simple case m(E) < ¥. For any  > 0, there is a countable open interval cover
{ } of E such that å ¥ n 1 (I ) n < m(E) + . Take O =  ¥ n 1 I , we see that O is open and O  E. Also,
I
=
=
n
n
we have
¥
m(O\E) = m(O) – m(E)  å m(I ) – m(E) < .
n
n 1
=
Here we use the assumption m(E) < ¥ and the countable subadditivity of m.
For the case m(E) = ¥, we write E =  ¥ E , where E = E  [–n, n]. This is a countable union of
=
n 1 n n
measurable sets of finite measure. By the above result there is an open set O such that O  E and
n n n

m*(O \E ) < . Take O =  ¥ O , then O is open and O  E. It remains to show m(O\E) <.
=
n n n n 1 n
2
Note that O\E   ¥ O \E , by countable subadditivity of m we have
n 1 n n
=
¥ ¥ 
m(O\E)  å m(O \E ) < å = .
n n n
n 1 n 1 2
=
=
Hence, we have proved that (1)  (2).
To show (2)  (4)
For any n  , let O be an open set such that O  E and m*(O \E) < 1/n. Take G =  ¥ O  G ,
=
n n n n 1 n 
then
1
m*(G\E)  m*(O \E) < .
n
n
Letting n  ¥, the result follows.
To show (4)  (1)
The existence of G guarantees E = G\(G\E) is measurable since both G and G\E are measurable
(G is Borel set and G\E is of measure zero).
Hence, (1), (2), (3), (4), (5) are equivalent.
To show (2)  (6) (with finiteness assumption m*(E) < ¥)
Let  > 0 be given. Let O be an open set such that O  E and m(O\E) < /2. Write O =  ¥ I to be
n 1 n
=
¥
a countable union of disjoint open intervals. By the countable additivity of m, m(O) = å n 1 (I ) n .
=
k
Let k be a positive integer such that å n 1 (I ) n > m(O) – /2. (The finiteness assumption has been
=
used here to guarantee that m(O) < ¥.)

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