Page 355 - DMTH401_REAL ANALYSIS

Unit 29: The Lebesgue Integral of Bounded Functions

 A function  : E  is said to be simple if there exists a , a ,...., a Î  and E , E ,...., E  E Notes
1 2 n 1 2 n
n
such that  = å i 1 a  i E . Note that here the E ’s are implicitly assumed to be measurable, so
=
i
i
a simple function shall always be measurable. We have another characterization of simple
functions:
 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 measurable set for all i = 1, 2,....., n.
2 n i
)
(a) A ò ( +  = A ò  + A ò  (Note that  +  ÎS (E) too by the vector space structure
0
(b) A ò a = a  for all a Î . (Note aÎS (E) again.)
A ò
0

(c) If a  a.e. on A then ò   A ò  .
A
(d) If  =  a.e. on A then ò  = A ò  .
A
(e) If  0 a.e. on A and ò  = , then  = 0 a.e. on A.
0
A
(f) A ò   A ò | |.(Note| | So (E) too.Why?)
Î


 Bounded Convergence Theorem Suppose m(E) < , and {f } is a sequence of measurable
n
functions defined and uniformly bounded on E by some constant M > 0, i.e.
|f |  M for all n on E.
n
29.5 Keyword
Bounded Convergence Theorem: Suppose m(E) < , and {f } is a sequence of measurable functions
n
defined and uniformly bounded on E by some constant M > 0, i.e.
|f |  M for all n on E.
n
29.6 Review Questions

1. Show that if A, B  E, A Ç B = 0 / and  Î S (E), then ò  = A ò  +  .
0 A B B ò
È
2. Show that if Î S (E) vanishes outside F, then ò  = ò  for any A E.
Ç
0 A A F
3. Show that if A  B  E and 0 Î S (E), then ò    .
0 A B ò
4. Find an example to show that the assumption m(E) <  cannot be dropped in the Bounded
Convergence Theorem.
5. Prove or disprove the following: Let E be of finite or infinite measure. If {f } is a sequence
n
of uniformly bounded measurable functions on E which vanishes outside a set of finite
measure and converges pointwisely to f Î B (E) a.e. on E, then lim f = f . (Compare
0 E ò n E ò
n
with the statement of the Bounded Convergence Theorem.)
Answers: Self Assessment

1. measurable set 2. finite measurable

3. measurable 4. Riemann integrable
5. |f |  M
n

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