Page 53 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes (ii) To show that f(x) is a continuous function. Note that if c',c'' F, then we have

c' 0. 2p 1 2p 2 2p ...
3
each p ,q 0 or =1
c'' 0. 2q 2q 2q ... i i
1 2 3
1
If c' c'' , then p = q , for 1 i n 1 and hence
3 n i i

1
f c' f c'' ...(i)
2 n

as n ,c' c'',f c' f c'' ,

Hence if c F and c is a sequence in F such that c c ,when n , then
0 n n 0
f c f c ,when n .
n 0
Now let x 0 [0,1] and let x be a sequence in [0,1] such that x n x as n .
0
n
Case I: Let x 0 F x 0 I,say a,b F c

x I and hence f x f x f a
n n
and hence f x f x as n .
n 0
Case II: Let x F. Now for each n such that x F,set x c and hence f x f x .
0 n n n n 0
c
If x n F,then an open interval I F .
(i) if x n x , then set c as the upper end point of I.
0
n
(ii) If x 0 x , then set c as the lower end point of I.
n
n
in any case f x n f x as n .
0
But the sequence x was any sequence satisfying the stated conditions.
n
f is a continuous function.
c
(iii) To show f(x) is not absolutely continuous. Note that f’(x) = 0 at each x F .
f' x exists and is zero on [0,1] and is summable on [0,1].
We know that for f(x) to be absolutely continuous, we must have

x
f x f' x dx f 0 .
0
Particularly, we must have

1
f 1 f 0 f' x dx.
0

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