Page 354 - DMTH401_REAL ANALYSIS

Real Analysis

Notes
f
= ò E\A f - + A ò |f - f| (by (e))
n
n

 ò + ò 2M (by our choice of N and that n  N)
E\A E\A
2m(E)
 m(E \A)
= + 2Mm(A)
2m(E)
 
 + 2M
2 4M
= .

Hence lim f exists (in ) and (5) holds.
E ò
n
n
(Alternatively when  > 0 is given, by Littlewood’s 3rd Principle we can choose a subset A of E
with m(A) < /4M such that {f } converges uniformly to f on E\A. Then choose N large enough
n
such that |f – f| < /2m(E) everywhere on E\A for all n  N, we see that whenever n  N, we
n
have (as in the above)
E ò f - E ò f < .
n
Hence lim f exists (in ) and (5) holds.)
E ò
n
n

Notes The first argument is just an adaptation of the proof of Littlewood’s 3rd Principal to
the present situation.

Self Assessment

Fill in the blanks:
1. A function  : E  is simple if and only if it takes only finitely many distinct values a ,
1
–1
a ,.... a and  {a } is a ......................... for all i = 1, 2,....., n.
2 n i
2. A function f : E   is said to vanish outside a set of ......................... if there exists a set A
with m(A) <  such that f vanishes outside A, i.e.
f = 0 on E\A
__
3. Let f be as in the above definition. Then ò f A ò = f for all A  E if and only if f is .......................
__ A
4. If f : [a, b]  is ......................................... on the closed and bounded interval [a, b], then
b
f Î B ([a, b]) and () f =  f, where the () and () represents Riemann integral and
( )ò
a ò
0 [a, b]
Lebesgue integral respectively.
5. Suppose m(E) < , and {f } is a sequence of measurable functions defined and uniformly
n
bounded on E by some constant M > 0, i.e. ..................... for all n on E.
29.4 Summary

 Recall that the characteristic function  for any set A is defined by
A
Î
1 if x A
 (x) =
A { 0 otherwise

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