Page 13 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Thus, the sequence <f > converges to f (x), a.e.
n
Using Fatou’s Lemma, we have

b b
f (x) dx Lim inf f (x) dx … (ii)
n n
a a
b b 1
Again Lim inf f (x) dx = Lim inf n f x f(x) dx
n
n a n a n
b 1 b
= Lim inf n f x dx f(x) dx
n n
a a
Putting t = x + (1/n), we get

b 1 b (1/n) b (1/n)
f x dx = f(t)dt f(x)dx
a n a (1/n) a (1/n)

[By the first property of definite integrals]
b b (1/n) b
Lim inf f (x) dx = Lim inf n f(x) dx f(x) dx
n n n
a a (1/n) a
b (1/n) a (1/n)
= Lim inf n f(x) dx f(x) dx … (iii)
n b a

Now extend the definition of f by assuming
f (x) = f (b), x [b, b + 1/n].

b (1/n) b (1/n) 1
f(x) dx = f(b) dx f(b)
b b n
1
Also f (a) f (x), for x a, a , therefore
n
a (1/n) a (1/n) 1
f(x) dx f(a) dx f(a)
a a n
a (1/n) 1
– f(x) dx f(a)
a n
b b (1/n) a (1/n)
(iii) Lim inf f (x) dx = Lim inf n f(b) dx f(x) dx
n
n a n b a
1 1
Lim inf n f(b) f(a) f(b) f(a)
n n n

Thus from (ii), we get
b
f (x) dx f (b) – f (a)
a
f (x) is integrable and hence finite a.e. thus f is differentiable a.e.

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