Page 23 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes

Notes

 This theorem enables us to define a new function (called variation function) say

x
V x V f , x [a,b] .
a
y x y
 If x > y in [a,b], then V f V f V f .
a a x
y
i.e. v(y) = v(x) + V f .
x
v x is an increasing function.

 If a c 1 c 2 ... c n b,then

b c 1 c 2 b
V f V f V f ... V f
a a c c
1 n
Corollary:

f BV[a,b] f BV[a,c],

f BV[c,b] for each c [a,b].

Theorem 5: If a function f of bounded variation in [a,b] is continuous at c [a,b], then the function
x
defined by v(x) = V f , is also continuous at x = c and vice versa.
a
Proof: Suppose f is continuous at x = c. Hence for arbitrary /2 0, we can find a such that
1
a c 1 x c or x c 1 f x f c /2 ...(i)

Also we know by remark (v) after the definition (2.1.3), for above , we can get a subdivision

P = a x ,x ,x ,...,x n c of [a,c]
1
0
2
c c
s.t. V f V f,P ...(ii)
a a 2
Now choosing positive min[ ,c x n 1 ], we get that for any x such that c x c , we also
1
have x n 1 x x .
n
c n 1
(ii) V f f x f x f x f x
a r r 1 n n 1 2
r 1
n 1
f x f x f x f x f x f x
r r 1 n n 1
r 1 2

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