Page 26 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 2: Functions of Bounded Variation

Notes
b b b
i.e. V h B.V f A.V g .
a a a
= a finite quantity.
Hence h(x) = f(x).g(x) is of bounded variation in [a,b].

(iv) First, we shall show that 1/g is of bounded variation, where g x 0, x [a,b].

Now, g x 0, x [a,b]

1 1
0, x [a,b].
g x

Again, we observe that

1 1 g x r g x r 1 1 g x g x
g x g x g x .g x 2 r r 1
r 1 r r r 1

n 1 1 1 1 n 1
g x g x
r 0 g x r 1 g x r 2 r 0 r r 1

b 1 1 b
V V g a finite quantity.
a g 2 a

1
Hence is of bounded variation in [a,b].
g

1
Now f and are of bounded variation in [a,b].
g

1
f. is of bounded variation in [a,b] [by case (iii)]
g
f
is of bounded variation in [a,b].
g

b b
(v) Do yourself. Note that V cf c V f .
a a

Notes

Since BV [a,b] is closed for all four algebraic operations, it is a linear space.

Theorem 7: Every absolutely continuous function f defined on [a,b] is of bounded variation.

Proof: Since f is absolutely continuous on [a,b]; for 1, a 0

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