Page 104 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

p
Unit 8: Bounded Linear Functional on the L -spaces

From (4) and (8) it finally follows that Notes
F = g .
q
This completes the proof of the theorem.

Approximation by Continuous Function

Theorem 6: If f is a bounded measurable function defined on [a, b], then for given > 0, a
continuous function g on [a, b], such that
f – g <
2
x
Proof: Let F (x) = f(t) dt where x [a, b].
a
x h x
Then |F (x + h) – F (x)| = f(t) dt f(t) dt
a a

x h x h
= f(t) dt f(t) dt
x x
Mh, where |f (x)| M, x [a, b].
Taking h < , and Mh < , we get
1
|x + h – x| < | F (x + h) – F (x) | <
1
F (x) is continuous on [a, b].

x h
Let G (x) = n f(t) dt : x [a, b] and n N;
n
x
1
then G (x) = n F x F(x) ( F (x) is continuous on [a, b]
n
n
G (x) is continuous on [a, b] n)
n
x
Again, since F (x) = f(t) dt, x [a, b].
a
F (x) = f (x) a.e. in [a, b].

F(x (1/n) F(x)
Now, Lim G (x) = Lim
n
n n 1/n
F(x h) F(x) 1
= Lim , h
h 0 h n
= F (x) = f (x) a.e. in [a, b]
2
and hence Lim G (x) f(x) = 0.
n
n

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