Page 46 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 4: Absolute Continuity

Notes
n
(ii) We have f b g b f a g a
r r r r
r 1
n
f b g b f b g a f b g a f a g a
r r r r r r r r
r 1

n
f b g b g a g a f b f a
r r r r r r
r 1

n n
f b g b g a g a f b f a
r r r r r r
r 1 r 1

n n
f b r g b r g a r g a r f b r f a .
r
r 1 r 1
Now every absolutely continuous function is bounded therefore f(x) and g(x) are bounded
in the closed interval [a,b].

Let f x K , g x K , x [a,b].
1 2
Then we have

n
f b g b f a g a K K K K ,
r r r r 1 2 1 2
r 1

n
Whenever b a .
r r
r 1

Setting K K *,
1 2
n
We have f b g b r f a g a r = *,
r
r
r 1
n
Whenever b r a r ,
r 1

where a b a b ... a b ;
1 1 2 2 n n
Product of two absolutely continuous functions is also absolutely continuous.

(iii) We have g x 0 x [a,b] ; therefore

g x ,where 0, x [a,b].

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