Page 39 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes then the bounded convergence theorem implies that
x x
F (x) dx = lim F (x) dx
h x
a a
x
1
= lim [F(x h) F(x)]dx
h 0 h
a
x h a h
1 1
= lim [F(x)dx F(x) dx
h 0 h h
x a
= F (x) – F (a)
x
= f(t) dt, by hypothesis
a
x x
 F(x) f(t)dt F(a) F(x) F(a) f(t)dt
a a
x
or [F (t) f(t)]dt 0, x
a
F (x) – f (x) = 0 a.e. in [a, b]

x
Hence F (x) = f(x) a.e. in [a, b] by the theorem, “If f is integrable on [a, b] and f(t) dt 0, x [a, b]
a
then f = 0 a.e. in [a, b]”.
Hence F (x) = f (x) a.e. in [a, b].
Hence the proof.

x
Theorem 3: If f is an integrable function on [a, b] and if F(x) = f(t) dt + F(a) then F (x) = f (x) a.e.
a
in [a, b].
Proof: Without loss of generality, we may assume that f (x) 0 x
Let us define a sequence {f } of functions
n
f : [a, b] R, where
n
f(x) if f(x) n,
f (x) =
n n if f(x) n
Clearly, each f is bounded and measurable function and so, by the theorem,
n
x
Let f be a bounded and measurable function defined on [a, b]. If F(x) = f(t) dt + F(a), then F (x)
a
= f(x) a.e. in [a, b]”, we have
x
d
f f (x) a.e.
dx n n
a

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