Page 372 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 3. The ......................................................... m * j,d [Y] of a bounded subset Y   is defined as
d
*
m [Y] = inf {å k n 1 Vol (A ) : k  , and A ’s are d-boxes with Y   k n 1 A }.
j,d
=
n
=
d
n
n
4. The construction above is due to Vitali, and hence the set Y is called a ............................
5. Let X, Y be metric spaces and let f: X  Y be a function. Then the oscillation (f, x) of f at
a point x  X is defined as (f, x) = lim diam [f(B(x, ))]. Clearly, f is ................................
0+
at x iff (f, x) = 0.
6. Let –¥  a < b  ¥ and let f: (a, b)   be a ........................................... Then, Y = {x  (a, b) : f
is discontinuous at x} is a countable set (possibly empty).
7. Let X be a metric space. Then the -algebra on X generated by the collection of all open
subsets of X is called the ............................... on X, and is denoted as (X) (or just , if X is
clear from the context).
8. If A  L is a ............................................. set, then A has a least upper bound in L . [Proof: If B
 
=   A L , then B is countable and hence L \B  Ø. The least element of L \B is the least



upper bound of A].
30.5 Summary
 Measure theory helps us to assign numbers to certain sets and functions – to a measurable
set we may assign its measure, and to an integrable function we may assign the value of its
integral. Lebesgue integration theory is a generalization and completion of Riemann
integration theory. In Lebesgue’s theory, we can assign numbers to more sets and more
functions than what is possible in Riemann’s theory. If we are asked to distinguish between
Riemann integration theory and Lebesgue integration theory by pointing out an essential
feature, the answer is perhaps the following.
d
d
(i) We say Y   is a discrete subset of  if for each y  Y, there is an open set U   d
such that U  Y = {y}. For example, {1/n: n  } is a discrete subset of .
d
d
(ii) A subset Y   is nowhere dense in  if int[ Y ] = Ø, or equivalently if for any
d
nonempty open set U   , there is a nonempty open set V  U such that V  Y =
0. For example, if f:    is a continuous map, then its graph G(f) := {(x, f(x)):x  }
2
is nowhere dense in  ( G(f) is closed and does not contain any open disc).
d
d
(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
2
example, Y =    is of first category in  since Y can be written as the countable
2
union Y =  r Y , where Y := {r}   is nowhere dense in  .
r
r
(iv) (The following definition can be extended by considering ordinal numbers, but we
d
consider only non-negative integers). For Y   and integer n  0, define the nth
(0)
d
(n)
derived set of Y inductively as Y = Y, Y (n+1) = {limit points of Y in  }. We say Y 
(n)
d
 has derived length n if Y  Ø and Y (n+1)  Ø; and we say Y has infinite derived
(n)
length if Y  Ø for every integer n  0. For example,  has infinite derived length
(since  = ), and {(1/m,1/n): m,n  } has derived length 2.
d
(v) We say A   is a d-box if A = Õ d I , where I’s are bounded intervals. The d-
=
j 1 j j
dimensional volume of a d-box A is Vol (A) = Õ d j 1 I . For example, Vol ([1, 4) 
d = j 3
[0,1/2]  (–1,3]) = 6.
*
(vi) The d-dimensional Jordan outer content m [Y] of a bounded subset Y   is defined
d
j,d
* k k
as m [Y] = inf {å Vol (A ) : k  , and A ’s are d-boxes with Y   n 1 A }.
=
j,d n 1 d n n = n
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