Page 52 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 4: Absolute Continuity

Notes
where f x n and f x n are the maximum and maximum values of f(x) in the interval [a ,b ].
n
n
Also note that n x x n b n a n

m * f E , being arbitrary.

m * f E 0 m f E 0.

Example: Give an example which is continuous but not absolutely continuous.

Solution: Consider the function f : F R, where F is the Cantor’s ternary set.

x
Let x F x x x x ... k ,x 0 or 2
1 2 3 3 k k
K 1
r 1
Define f x k , where r x .
2 k k 2 k
K 1
= 0. r , r , r .....
1 2 3
This function is continuous but not absolutely continuous.
(i) Note that this function is constant on each interval contained in the complement of the
Cantor’s ternary set.

c
For, let (a,b) be one of the countable open intervals contained in F . Then in ternary
notation,
a = 0.a a ...a 0 2 2 2
1 2 n–1
and b = 0.a a ...a 2 0 0 0,
1 2 n-1
where a = 0 or 2, for i n 1.
i

a
f a 0.r ,r ,...,r 0 1 1 1 1 ...,where r i i ,
1
n 1
2
2
f b 0.r ,r ,...,r 1 0 0 0 0 ...
1 2 n 1
But in binary notation
0.r ,r ,...,r 0 1 1 1 1 ... 0.r ,r ,...,r 1 0 0 0 0 ...
2
n 1
1
2
n 1
1
f a f b .
Thus, we extend the function f overall of the set [0,1] instead of F by defining
c
f x f b , x a,b F . Thus, the Cantor’s function is defined over [0,1] and maps it
onto [0,1].
It is clearly a non-decreasing function.

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