Page 29 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
since f is finite in [a,b] Now g b ,g a h b ,h a are finite numbers.
n 1
f x h x
r 1 r
r 0
b
V f .
a
f is a function of bounded variation. Alternatively, since g (x) and h(x) are both non-decreasing,
so by theorem 3, g(x) – h(x) and hence f(x) is of bounded variation.
Corollary: A continuous function is of bounded variation iff it can be expressed are as a difference
of two continuous monotonically increasing functions. It follows from the results of Theorems
5 and 8.

Theorem 9: An indefinite integral is a function of bounded variation, i.e. if f L[a,b] and F x is

x
indefinite integral of f x i.e. F (x) = f t dt, then F BV[a,b].Also show that
a

x
b
V f f .
a
a
Proof: Since f L[a,b], also f L[a,b].

Let P = x : i 0,1,2,...,n be a subdivision of the interval [a,b]. Then
i

x x
n n i i 1
F x i F x i 1 f f
r 0 i 1 a a

x x
n i n i
f f
i 1 x i 1 x
i 1 i 1
b
f .
a
b
b
f BV[a,b] and V f,p f .
a
a
Further above result is true for any subdivision of P of [a,b]. Therefore taking supremum, we get

b b
f f .
a a

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