Page 373 - DMTH401_REAL ANALYSIS

Unit 30: Riemann's and Lebesgue

d
(vii) The d-dimensional Lebesgue outer measure m * L,d [Y] of an arbitrary set Y   is Notes
defined as m * L,d [Y] = inf {å ¥ n 1 Vol (A ) : A ’s are d-boxes with Y   ¥ n 1 A }.
n
=
=
n
d
n
 If f : [a, b]   is a function and P = {a = a  a  … a  a = b} is a partition of [a, b],
0 1 n – 1 n
b
b
b
let V (f, P) = å n i 1 |f(a ) – f(a i – 1 )|. Define the total variation of f as V (f) = sup{ V (f, P) : P
a
a
a
=
i
b
is a partition of [a, b]}. We say f is of bounded variation if V (f) < ¥. It is easy to see that
a
if f is of bounded variation, then f is bounded ( if x  [a, b], take P = {a  x  b} to see that
b
|f(x) – f(a)|  V (f)).
a
30.6 Keywords
Riemann Integration Theory: Riemann integration theory  finiteness.
Lebesgue Integration Theory: Lebesgue integration theory  countable infiniteness.
Baire Category Theorem: Let (X, ) be a complete metric space and let U  X be open and dense
n
in X for n  . Then,  ¥ n 1 U is also dense in X. In particular,  ¥ n 1 U  Ø.
=
=
n
n
Lebesgue’s Differentiation Theorem: Let –¥  a < b  ¥, let f : (a, b)   be a monotone function
and let Y = {x  (a, b) : f is not differentiable at x}. Then m * L,1 [Y] = 0.
Borel -algebra: If X is a separable metric space, then any base or subbase for the topology of X
will generate the Borel -algebra (X).
Well-ordering Principle: Well-ordering principle (equivalent to the axiom of choice): Any
non-empty set admits a well-ordering.
30.7 Review Questions
1. If f, g : [a, b]   are of bounded variation, then fg is of bounded variation. [Hint: Let M > 0
be such that |f|,|g|  M. Now, subtracting and adding the term f(a )g(a ), note that
i i – 1
b
|(fg)(a ) – (fg)(a )|  |f(a )||g(a ) – g(a )| + |f(a ) – f(a )||g(a )| and hence V (fg)
i i – 1 i i i – 1 i i – 1 i – 1 a
b
b
 M( V (f) + V (g)).]
a a
b
c
b
2. If f : [a, b ]   is a function and c  [a, b ], then V (f) = V (f) + V (f). [Hint: If P is a partition
a a c 1
c
b
b
b
of [a, c] and P is a partition of [c, b], then V (f, P ) + V (f, P ) = V (f, P  P )  V (f).
2 a 1 c 2 a 1 2 a
Conversely, if P is a partition of [a, b], first refine it by inserting c and then divide into
b
b
b
c
c
partitions P of [a, c] and P of [c, b]. Check that V (f, P)  V (f, P ) + V (f, P )  V (f) + V (f).]
1 2 a a 1 c 2 a c
3. Let f: [a, b]   be a bounded function. If f is either monotone or of bounded variation,
then f is Riemann integrable.
4. If f, g; [a,b]   are Riemann integrable, then h : = max{f, g} is also Riemann integrable.
[Hint: The set of discontinuities of h is contained in {x : f is not continuous at x}  {x : g is not
continuous at x}.]
5. If  is a -algebra on a set X show that
(i) A\B, A B   if A, B  ,
(ii)  ¥ n 1 A   if A ,A ,… .
=
1
2
n
6. Let  = {A   : A is a countable (possibly finite or empty) union of d-boxes}. Is  a
d
d
-algebra on  ? [Hint: Let d = 1. Consider  and \, or the middle-third Cantor set and
its complement.]
d
7. Are the following -algebras on  :  = {A   : A or  \A is open in  } and  = {A   :
d
d
d
d
1 2
d
d
A or  \ A is dense in  }?
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