Page 310 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Next, we prove it is always possible to have a disjoint collection. Since {I } is a countable
x xV
collection, we can enumerate the open intervals as I = (a , b ), I = (a ,b ),.... For each n  ,
1 1 1 2 2 2
define
 = inf{ x   : x  a and (x, b )  V}
n n n
and
 = sup{ x   : x  b and (a , x)  V}.
n n n
Then { ( , ) } is a disjoint collection of open intervals with union V.
n n
Theorem 1 (Lindelof’s Theorem): Let C be a collection of open subsets of . Then there is a
countable sub-collection {O } i i of C such that

¥
 O =  O i
=
O C i 1

Proof: Let U =  o C O. For any x  U there is O  C with x  O. Take an open interval I with


x
rational endpoints such that x  I  O. Then U =  I is a countable union of open intervals.

x x U x
Replace I by the set O  C which contains it, the result follows.
x
26.2 Lebesgue Outer Measure
As in the Archimedean idea of finding area of a circle (approximated polygons), we define the
Lebesgue outer measure m*: ()  [0, ¥] by
m*(A) = { ¥ (I ) : A   I and each I being open interval in  } .
¥
inf å 
k 1 k k 1 k k
=
=
Notes By Lindelof’s Theorem, the countability of the covering is not important here.

Here are some basic properties of Lebesgue outer measure, all of them can be proved easily by
the definition of m*.
(i) m*(A) = 0 if A is at most countable.

(ii) m* is monotonic, i.e. m*(A)  m*(B) whenever A  B.
(iii) m*(A) = inf {m*(O): A  O and O is open}. (Hint: it suffices to prove m*(A)  R.H.S., which
is equivalent to m*(A) +> R.H.S. for any  > 0.)

(iv) m*(A + x) = m*(A) for all x  . (Translation-invariant)
(v) m* ( k A )  å ¥ k 1 m* (A ). (Countable subadditivity)

K
=
k
(vi) If m*(A) = 0, then m*(A  B) = m*(B) and m*(B\A) = m*(B) for all B  R.
(vii) If m*(A B) = 0, then m*(A) = m*(B).

Notes In (v), even if A ’s are disjoint, the equality may not hold.
k
Theorem 2: For any interval I  R, m*(I) = (I).

Proof: We first assume I = [a, b] is a closed and bounded interval. Consider the countable open
interval cover {(a – , b + )} of I, we have m*(I)  b – a + 2. Since  > 0 is arbitrary, m*(I)  b – a.

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