Page 28 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 2: Functions of Bounded Variation

Again, if x<y in [a,b], then as above Notes
y
v y v x V f f y f x f y f x
x

v y f y v x f x v f y v f x

v f is also a non-decreasing function on [a,b].

Thus (i) shows that f is expressible as a difference of two monotonically non-decreasing functions.
Case II. Set g (x) and h (x) be increasing functions such that f(x) = g(x) – h(x).
Divide the closed interval [a,b] by means of points

a = x < x < x <...< x =b.
0 1 2 n
n 1
Let V f x f x
r 1 r
r 0
Now, we have that
f x f x g x h x g x h x
r 1 r r 1 r 1 r r
g x g x h x h x
r 1 r r r 1
g x r 1 g x r h x r h x r 1
g x g x h x h x
r 1 r r 1 r
Now, g(x) and h(x) are monotonically increasing functions, so that g x r 1 g x r 0

and h x h x 0
r 1 r
g x g x g x g x
r 1 r r 1 r

and h x r 1 h x r h x r 1 h x .
r
Hence f x r 1 f x r g x r 1 g x r h x r 1 h x r

n 1 n 1 n 1
f x f x g x g x h x h x
r 1 r r 1 r r 1 r
r 0 r 0 r 0
n 1
Now g x r 1 g x r g x 1 g x 0 g x 2 g x 1 ..... .... g x n g x n 1
r 0

g x n g x 0
g b g a  x b,x a
n 0
n 1
Similarly, h x h x h b h a .
r 1 r
r 0
n 1
Hence f x r 1 f x r g b g a h b h a .
r 0

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