Page 22 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 2: Functions of Bounded Variation

Notes
f x – f x i 1 f x – f x i 1 f x – f x i 1
1
2
i
1
i
2
i
f b – f a f b – f a
1
2
2
1
as f and f are monotonically increasing.
1 2
b
V f f b f b f a f a , which is a finite quantity.
2
1
1
2
a
b
V f and hence f is of bounded variation.
a
Theorem 4: If f BV [a,b] and c a,b , then f BV [a,c] and f BV [c,b] . Also
b c b
V f V f V f
a a c
Proof: Since f BV[a,b] and [a,c] [a,b] we get
c b
V f V f
a a
f BV [a,c] and similarly f BV [c,b] .
Now if P and P are any subdivisions of [a,c] and [c,b] respectively, then P P P is a
1 2 1 2
subdivision of [a,b].
c b b b
V f,P 1 V f,P 2 V f,P V f .
a c a a
But P and P are any subdivisions. So taking supremums on P and P , we get
1 2 1 2
c b b
V f V f V f . ...(1)
a c a
Now let P a x ,x ,x ,...,x b be a subdivision of [a,b] and c x ,x
0 1 2 n r 1 r
P 1 x ,x ,x ,...,x ,c and
2
0
1
r 1
P c,x ,x ,x ,...,x are the subdivisions of [a,c] and [c,b] respectively.
2 r r 1 r 2 n
b r 1 n
Now V f,P f x i f x i 1 f x r f x r 1 f x i f x i 1
a
i 1 i r 1
r 1 n
f x i f x i 1 f x r f c f c f x r 1 f x i f x i 1
i 1 i r 1
r 1 n
f x i f x i 1 f c f x r 1 f x r f c f x i f x i 1
i 1 i r 1
c b c b
V f,P 1 V f,P 2 V f V f
a c a c
b c b
(i) and (ii) V f V f V f . ...(ii)
a a c

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