Page 41 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes b
But F (x) = f (t) dt F(a)
a
b
F(b) – F(a) = f (x) dx
a
Therefore (2) becomes
b b
F (x) dx f(x) dx … (3)
a a
From (1) and (3), we get
b b
F (x) dx = f(x) dx
a a
b b
F (x) dx f(x) dx = 0
a a
b
F (x) f(x) dx = 0
a

since F (x) – f(x) 0 a.e., which gives that
F (x) – f(x) = 0 a.e. and
so F (x) = f(x) a.e.

3.2 Summary

 If f is an integrable function on [a, b] then f is integrable on any interval [a, x] [a, b]. The
function F given by

x
F (x) = f(t) dt c ,
a
where c is a constant, called the indefinite integral of F.

x
 Let f be an integrable on [a, b]. If f(t) dt 0 x [a,b]then f = 0 a.e. in [a, b].
a

3.3 Keyword

Differentiation of an Integral: If f is an integrable function on [a, b] then f is integrable on any
interval [a, x] [a, b]. The function F given by

x
F (x) = f(t) dt c ,
a
where c is a constant, called the indefinite integral of f.

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