Page 48 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 4: Absolute Continuity

Notes
n n
f b f a = f b f a f a f a
r r r r
r 1 r 1
n
f b f a f a f a
r r
r 1
Now taking supremum over all collections of P of [a , b ] for i = 2,...,n, we get
i i i
n b
i
V f .
a i
r 1
b i a i b i
But V f V f V f
a a a
i
b b a
i i i
V f V f V f
a a a
i
b i
V f v b i v a i
a i

n
v b v a <
i i
i 1
v x is absolutely continuous.

Theorem 4: A necessary and sufficient condition that a function should be an indefinite integral
is that it should be absolutely continuous.
Proof: Condition is sufficient.
Let f(x) be an absolutely continuous function over the closed interval [a,b].

Therefore f is of bounded variation and hence we can express f(x) as
f(x) = f (x) – f (x)
1 2
where f (x) and f (x) are monotonically increasing functions and hence both are differentiable.
1 2
n n
v b v a < whenever b a
r r r r
r 1 r 1
f is also absolutely continuous on [a,b].
Case II: Given f is absolutely continuous on [a,b].

for a given 0, a 0 s.t.

n
f b f a < , ...(i)
i i
i 1

LOVELY PROFESSIONAL UNIVERSITY 41