Page 359 - DMTH401_REAL ANALYSIS

Unit 30: Riemann's and Lebesgue

integer n  0. For example,  has infinite derived length (since  = ), and {(1/m,1/n): Notes
m,n  } has derived length 2.
I ,
(v) We say A   is a d-box if A = p d j 1 j where I’s are bounded intervals. The d-dimensional
d
=
j
volume of a d-box A is Vol (A) = p d I . For example, Vol ([1, 4)  [0,1/2]  (–1,3]) = 6.
d j 1 j 3
=
*
d
(vi) The d-dimensional Jordan outer content m [Y] of a bounded subset Y   is defined as
j,d
*
m [Y] = {å k n 1 infVold(An) : k  , and A ’s are d-boxes with Y   k n 1 A }.
j,d
n
=
=
n
d
(vii) The d-dimensional Lebesgue outer measure m * L,d [Y] of an arbitrary set Y   is defined as
m * [Y] = inf {å ¥ Vol (A ) : A ’s are d-boxes with Y   ¥ n 1 A }.
=
L,d n 1 d n n = n
We have that m * L,d [Y]  m * J,d [Y] for any bounded set Y   , and m * L,d [A] = m * J,d [A] = Vol (A) for
d
d
d
any d-box A   .
Proof: Any finite union  k n 1 A of d-boxes can be extended to an infinite union  ¥ n 1 A of d-boxes
n
n
=
=
without changing the total volume by taking A ’s to be singletons for n > k. This observation
n
yields that m * L,d [Y]  m * J,d [Y]. It is easy to see m * J,d [A] = Vol (A) if A is a d-box. It remains to
d
show m * L,d [A]  Vol (A) when A is a d-box. First suppose A is closed. Then A is compact by
d
Heine-Borel. Let  > 0 and let A ,A ,…   be d-boxes such that A   ¥ n 1 A and å ¥ n 1 Vol (A )
d
n
=
=
1
d
2
n
n
< m * L,d [A] + . For each n  , let B be an open d-box with A  B and Vol (B ) < Vol (A ) + /2 .
n
d
n
n
n
d
n
Then {B : n  } is an open cover for the compact set A. Extracting a finite subcover, we have
n
n
Vol (A)  å k Vol (B )  å ¥ (Vol (A ) + /2 ) < m * [A] + 2. Thus m * [A] = Vol (A) for closed
=
=
d n 1 d n n 1 d n L,d L,d d
d-boxes. Now if B is an arbitrary d-box and  > 0, then there is a closed d-box A  B with Vol (B)
d
–  < Vol (A) = m * [A]  m * [B].
d L,d L,d
Other basic properties of Lebesgue outer measure and Jordan outer content are given below.
(i) m * L,d [Ø] = 0.
*
(ii) [Monotonicity] m L,d [X]  m * L,d [Y] if X  Y   .
d
d
(iii) [Translation-invariance] m * L,d [Y + x] = m * L,d [Y] for every Y   and every x   .
d
(iv) [Countable subadditivity] If Y ,Y ,…  and Y =  ¥ Y , then m * [Y]  å ¥ m * [Y ].
d
=
=
1 2 n 1 n L,d n 1 L,d n
d
(v) m * L,d [Y] = 0 for every countable set Y   .
d
(vi) Forany Y   , we have m * L,d [Y]
= {å ¥ n 1 infVold(An) : A ’s are closed d-boxes with Y   ¥ n 1 A }
=
=
n
n
= {å ¥ n 1 infVold(An) : A ’s are open d-boxes with Y   ¥ n 1 A }.
=
=
n
n
(vii) For any Y   , we have m * L,d [Y] = m * L,d inf{[U] : Y  U and U is open in  }.
d
d
(viii) m * L,d [ ] = ¥.
d
(ix) If X,Y   are such that dist(X,Y) := inf{||x – y|| : x  X, y  Y} > 0, then m * L,d [X  Y] =
d
m * [X] + m * [Y].
L,d L,d
Proof: (i), (ii) and (iii) are clear. To prove (iv), without loss of generality we may assume
å ¥ n 1 m * L,d [Y ] < ¥. Given  > 0, there exist d-boxes A(n,k) such that Y   ¥ k 1 A(n,k) and
=
=
n
n
¥
n
å ¥ k 1 Vol (A(n,k)) < m * L,d [Y ] +/2 . Then Y   n 1  ¥ k 1 A(n,k) and we have the estimate
=
=
=
d
n
n
å ¥ n 1 å ¥ k 1 Vol (A(n,k))  å ¥ n 1 ( m * L,d [Y ] + /2 ) = ( å ¥ k 1 m * L,d [Y ]) + .
=
=
=
=
d
n
n
Now (v) follows from (iv) since singletons have Lebesgue outer measure zero (or we can see it
directly by noting that singletons are d-boxes with zero volume). The first part of (vi) is clear
since any d-box and its closure have equal volume. To get the second part, note that if A ,A ,…
1 2
are d-boxes and  > 0, there exist open d-boxes B ,B ,… such that A  B and Vol (B ) < Vol (A )
1 2 n n d n d n
n
+ /2 . We may deduce (vii) using part (vi).
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