Page 314 - DMTH401_REAL ANALYSIS

Real Analysis

Notes By the definition of m*(A), there is a countable open interval cover { } of A with
I
n n
¥
m*(A) + > å  (I ).
n
n 1
=
²
{ } { }

Let I = I  [–¥, a] and I = I  (a, ¥), then I  n , I ² n are, respectively, interval covers of A and
n
n
1
n
n
A (note that they may not be open interval covers). Then
2
¥ ¥ ¥
å  (I ) = å  (I ) + å  (I )
²

=
n 1 n n 1 n n 1 n
=
=
¥ ¥
²

= å m* (I ) + å m* (I ) ( m* =  for intervals)
n
n
=
=
n 1 n 1
( ) ( )
¥
¥
 m*  I  n + m*  I ² n ( countable subadditivity)
=
n 1 n 1
=
 m*(A ) + m*(A ) ( monotonicity)
1 2
¥
So, m*(A) +  > å n 1 (I ) n  m*(A ) + m*(A ) for all  > 0. Letting   0, m*(A)  m*(A ) + m*(A ).
=
1
2
1
2
This proved that (a, ¥)  M.
Notes Since M is a -algebra, (–¥, a]  M and (–¥, a) =  ¥ (–¥,a – 1/n]  M. It follows
n 1
=
that (a, b)  M since (a, b) = (–¥, b)  (a, ¥). As M is a -algebra containing all open
intervals, it must contain all open sets (recall that every open set is countable union of
open intervals by Proposition). Therefore, M contains all Borel sets.
Proposition: M is translation invariant: for all x  , E  M implies E +x  M.
Proof: For all A  R, we have
m*(A) = m*(A – x)
= m* ((A – x)  E) + m* ((A – x)  E )
c
c
= m* (((A – x)  E) + x) + m* (((A – x)  E ) + x)
= m*(A  (E + x)) + m*(A  (E + x) )
c
Notes Let E   be given. Then the following statements are equivalent.
1. E is measurable.
2. For any  > 0, there is an open set O  E such that m*(O\E) <.
3. For any  > 0, there is a closed set F  E such that m*(E \F)<.
4. There is a G  G such that E  G and m*(G\E) = 0.

5. There is a F  F such that E  F and m*(E\F) = 0.

Assume m*(E) < ¥, the above statements are equivalent to
6. For any  > 0, there is a finite union U of open intervals such that m*(U  E) < .

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