Page 40 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 3: Differentiation of an Integral

Also, f – f > 0 n, and therefore, the function G defined by Notes
n n
x
G (x) (f f )
n n
a
is an increasing function of x, which must have a derivative almost everywhere by Lebesgue
theorem and clearly, this derivative must be non-negative.

x
Since G (x) = (f f )
n
n
a
x x
= f(t) dt f (t) dt
n
a a

x x
f(t) dt = G (x) f (t) dt
n n
a a
Now the relation
x
F (x) = f(t) dt F(a) becomes
a

x
F (x) = G (x) f (t) dt F(a) ,
n
n
a
F (x) = G (x) f (x) a.e.
n n
f (x) a.e. n.
n
since n is arbitrary, we have
F (x) f(x) a.e.
b b
F (x) dx f (x) dx … (1)
a a
Also by the Lebesgue’s theorem, i.e. “Let f be an increasing real-valued function defined on
[a, b].
Then f is differentiable a.e. and the derivative f is measurable.

b
and f (x) dx f (b) – f (a)”, we have
a

b
F (x) dx F (b) – F (a) … (2)
a

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