Page 358 - DMTH401_REAL ANALYSIS

Real Analysis

Notes not be any consistent way to assign numbers to  ¥ n 1 A , å ¥ n 1 n n ¥ f , sup{f : n  }, etc.
f , lim
=
n
=
n
n
even if we can assign numbers to the sets A , A ,…, and functions f , f …. Lebesgue integration
1 2 1 2
theory rectifies this disadvantage to a large extent.
In Riemann integration theory, we proceed by considering a partition of the domain of a function,
where as in Lebesgue integration theory, we proceed by considering a partition of the range of
the function – this is observed as another difference. Moreover, while Riemann’s theory is
restricted to the Euclidean space, the ideas involved in Lebesgue’s theory are applicable to more
general spaces, yielding an abstract measure theory. This abstract measure theory intersects
with many branches of mathematics and is very useful. There is even a philosophy that measures
are easier to deal with than sets.
30.2 Small Subsets of  d

It is possible to think about many mathematical notions expressing in some sense the idea that
d
d
a subset Y   is a small set (or a big set) with respect to  . We will discuss this a little as a
warm-up. We will also use this opportunity to introduce Lebesgue outer measure.
d
Suppose you have a certain notion of smallness or bigness for a subset of  . Then there are some
natural questions. Two sample questions are:
d
d
1. If Y   is big, is  \Y small?
d
2. If Y , Y ,…   are small, is  ¥ n 1 Y small?
1 2 = n
d
For instance, consider the following two elementary notions. Saying that Y   is unbounded
d
is one way of saying Y is big, and saying that Y   is a finite set is one way of saying Y is small.
Note that the complement of an unbounded set can also be unbounded and a countable union of
finite sets need not be finite. So here we have negative answers to the above two questions.

Task Find an uncountable collection {Y :   I} of subsets of  such that Y ’s are pairwise
 
disjoint, and each Y is bounded neither above nor below.

To discuss some other notions of smallness, we introduce a few definitions.
Definitions:
d
d
d
(i) We say Y   is a discrete subset of  if for each y  Y, there is an open set U   such
that U  Y = {y}. For example, {1/n: n  } is a discrete subset of .
(ii) A subset Y   is nowhere dense in  if int[ Y ] = Ø, or equivalently if for any non-empty
d
d
open set U   , there is a nonempty open set V  U such that V  Y = 0. For example, if
d
f:    is a continuous map, then its graph G(f) := {(x, f(x)):x  } is nowhere dense in  2
( G(f) is closed and does not contain any open disc).
(iii) A subset Y   is of first category in  if Y can be written as a countable union of
d
d
nowhere dense subsets of  ; otherwise, Y is said to be of second category in  . For
d
d
2
example, Y =    is of first category in  since Y can be written as the countable union
Y =  r Y , where Y := {r}   is nowhere dense in  .
2
r
r
(iv) (The following definition can be extended by considering ordinal numbers, but we consider
d
only non-negative integers). For Y   and integer n  0, define the nth derived set of Y
d
(0)
inductively as Y = Y, Y (n+1) = {limit points of Y in  }. We say Y   has derived length
d
(n)
n if Y  Ø and Y (n+1)  Ø; and we say Y has infinite derived length if Y  Ø for every
(n)
(n)
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