Page 44 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 4: Absolute Continuity

Notes

Notes

1. If a function satisfied f b r f a r , even then it is absolutely continuous.

n
2. The condition b r a r means that total length of all the intervals must be less
r 1
than .

4.1.2 Theorems and Solved Examples

Theorem 1: Every absolutely continuous function f defined on [a,b] is of bounded variation.

Proof: Since f is absolutely continuous on [a,b]; for 1, a 0 such that

n
f b f a 1,
i i
r 1

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whenever b a ,
i i
r 1

and a a 1 b 1 a 2 b 2 ... a n b n b.
Now consider another subdivision of [a,b] or say refinement of P by adjoining some additional
points to P in such a way that all the intervals can be divided into r parts each of total length less
than .
Let the r-sub-intervals be c ,c , c ,c ,..., c ,c such that
1
0
1
r
r 1
2
a c ,c r b and c k+1 c k , k 0,1,2,..., r 1
0
Obviously, f x i 1 f x i 1,
i
where x ,x c ,c
i 1 i k k 1
c
or k 1 1,
V f
c
k
b c 1 c 2 c r
Hence V f V f V f ... V f 1 1 ... 1 r finite quantity.
a c c c
0 1 r 1
Hence f is of bounded variation.

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