Page 49 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
for every finite collection P a ,b , i 1,2,...,n of pairwise disjoint sub-intervals of [a,b] such
i i
n
that b i a .
i
i 1

Now, let P i x i k 1 ,bx i ki ,k 1,...,m i be a finite collection of non-overlapping intervals of the
interval [a ,b ].
i i
Then the collection x k i 1 ,bx ki i : i 1,2,..., ,k 1,...,m is a finite collection of non-overlapping
n
i
sub-intervals of [a,b] such that
n m i n
x i x i b a .
k k 1 i i
i 1 k 1 i 1

n m i
and hence by (i), f x i k f x i k 1 .
i 1 k 1

'
'
Hence f’(x) exists and f' x f x f x
1 2
b
f x f b f b f a f a ,
1 2 1 2
a
f x is integrable also.
Now let F(x) be an definite integral of f’(x) i.e.

x
F(x) = F(a) + f t dt, x [a,b] ...(ii)
a
Using fundamental theorem of integral calculus,
We get

F’(x) = f’(x)
or F(x) = f(x) + constant (say c) ...(iii)
From (ii), we have F(a) = f(a),

Using this in (iii), we get c = 0 and hence F (x) = f(x).
Thus every absolutely continuous function f(x) is an indefinite integral of its own derivative.
Condition is necessary: Let f(x) be an indefinite integral of f(x) defined on the closed interval
[a,b], so that
x
F x f t dt f a , x [a,b] and f(x) is integrable over [a,b].
a

Corresponding to arbitrary small 0, let >0 be such that if m(A) ,then f ,
A

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