Page 360 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Now we prove (ix). From countable subadditivity, we have m * L,d [X  Y]  m * L,d [X] + m * L,d [Y]. To
prove the other inequality, we may assume m * L,d [X  Y] < ¥. Let  = dist(X,Y). Given  > 0, find
d-boxes A ,A ,… such that X  Y   ¥ A and å ¥ Vol (A ) < m * [X  Y] + . By partitioning
=
=
1 2 n 1 n n 1 d n L,d
the d-boxes into smaller d-boxes and throwing away the unnecessary ones, we may assume that
diam[A ] <  and (X  Y)  A  Ø for every n  . Let = {n  : X  A  Ø} and  = {n  :
n n n
Y  A  Ø}. Then  =    is a disjoint union, X   n A , and Y   n A . Hence m * L,d [X] +
n
n
n
m * [Y]  å n Vol (A ) + å n Vol (A ) = å ¥ Vol (A ) < m * [X  Y] + .
L,d d n d n n 1 d n L,d
=
(i) m * J,d [Ø] = 0.
(ii) [Monotonicity] m * J,d [X]  m * L,d [Y] if X  Y are bounded subsets of  .
d
(iii) [Translation-invariance] m * L,d [ Y + x] = m * L,d [Y] for every bounded set Y   and every
d
x   .
d
(iv) [Finite subadditivity] If X,Y   are bounded subsets, then m * J,d [X  Y]  m * J,d [X] + m * J,d [Y].
d
d
(v) m * J,d [Y] = 0 for every finite set Y   .
(vi) For any bounded set Y   , we have m * J,d [Y]
d
= inf {å k n 1 Vol (A ) : k  , and A ’s are closed d-boxes with Y   k n 1 A }
=
=
n
n
d
n
= inf {å k n 1 Vol (A ) : k  N, and A ’s are open d-boxes with Y   k n 1 A }
=
=
n
n
d
n
= inf {å k n 1 Vol (A ) : k  N, and A ’s are pairwise disjoint d-boxes with Y   k n 1 A }.
=
=
d
n
n
n
d
(vii) If X,Y   are bounded sets with dist(X,Y) := inf{||x – y||: x  X, y  Y} > 0, then m * J,d [X  Y]
= m * J,d [X] + m * J,d [Y].
d
(viii) For any bounded set Y   , we have m * J,d [ Y ] = m * J,d [Y].
Proof: To prove (viii), use the first expression for m * J,d [Y] in (vi) and note that a finite union of
closed sets is closed.
*
d
Example: Let Y =   [0, 1] . Note that m * L,d [Y] = 0  1 = m L,d [ Y ]. But we have m * J,d [Y] =
d
m * J,d [ Y ] = 1. So the Jordan outer content of a bounded countable set need, not be zero. This
example also shows that m * L,d [Y] < m * J,d [Y] is possible for a bounded set, and that the Jordan outer
content does not satisfy countable subadditivity for bounded sets (since the Jordan outer content
of a singleton is zero). If X = [0, 1] \Y, then m * J,d [X] = 1 since X = [0, 1] and hence m * J,d [X] + m * J,d [Y]
d
d
= 2  1 = m * J,d [X  Y].
Some ways of saying that Y   is a small set:
d
(i) Y is a countable set.
(ii) Y is a discrete subset of  .
d
(iii) Y is contained in a vector subspace of  of dimension  d – 1.
d
(iv) Y is nowhere dense in  .
d
d
(v) Y is of first category in  .
(vi) Y has finite derived length.
(vii) Y is a bounded set with m * J,d [Y] = 0.
(viii) m * L,d [Y] = 0.
It is good to investigate various possible implications between pairs of notions given above.
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