Page 25 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
f x f x g x g x
r 1 r r 1 r
f x f x g x g x .
r 1 r r 1 r
n 1 n 1 n 1
h x h x h x h x g x g x .
r 1 r r 1 r r 1 r
r 0 r 0 r 0
b b b
or V h V f V g .
a a a
Now by hypothesis, f, g are functions of bounded variations.

b b
V f andV g are finite.
a a
b
V h a finite quantity.
a
Hence h = f + g is of bounded variation in [a,b].
(ii) Let h = f – g. Then as above,

h x r 1 h x r f x r 1 f x r g x r 1 g x r .

b b b
V h V f V g
a a a
b
V h a finite quantity.
a
Hence h = f – g is of bounded variation in [a,b].

(iii) Let h(x) = f(x).g(x). Then
h x h x f x .g x f x .g x
r 1 r r 1 r 1 r r
f x .g x f x g x f x g x f x g x
r 1 r 1 r r 1 r r 1 r r

g x r 1 f x r 1 f x r f x r g x r 1 g x r .

g x f x f x f x . g x g x .
r 1 r 1 r r r 1 r
Let A = sup f x : x [a,b] ,

B = sup g x : x [a,b] ,

h x h x B. f x f x A. g x g x .
r 1 r r 1 r r 1 r
n 1 n 1 n 1
h x r 1 h x r B. h x r 1 h x r A g x r 1 g x r .
r 0 r 0 r 0

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