Page 27 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes n
s.t. f b f a 1,
i i
i 1
n
whenever b i a i ,
i 1
and a a 1 b 1 a 2 b 2 ... a n b n b.

Now consider another subdivision of [a,b] or say refinement of P by adjoining some additional
points to P in such a way that all the intervals can be divided into r parts each of total length less
than .
Let the r sub-intervals be [c ,c ], [c ,c ],...,[c ,c ] such that
0 1 1 2 r-1 r
a = c , c = b and (c –c ) < , K 0,1,2,...,(r 1)
0 r k+1 k

Obviously, f x i 1 – f x i 1 1,where x ,x . [c ,c k 1 ]
k
i 1
i
i
c k 1
or V f 1, [Using (i)]
c k
b c 1 c 2 c r
Hence V f V f V f ... V f 1 1 1 ... 1 r finite quantity.
a c c 1 c
0 r 1
Hence, f is of bounded variation.

Notes

Converse of above theorem is not necessarily true. These exists functions of bounded
variation but not absolutely continuous.

Theorem 8: Jordan Decomposition Theorem
A function f is of bounded variation, if and only if it can be expressed as a difference of two
monotonic functions both non-decreasing.
Proof: Let f be the function of f :[a,b] R.

Case I. f BV[a,b]. Then we can write

f = v – (v – f), ...(i)
so that f x v x v x f x ,x [a,b].

Now if x, y [a, b] such that x < y, then by the remark (ii) of theorem 4, we get
y x y
V f V f V f .
a a x
y
v y v x V f 0
x

v x v y and hence v is a non-decreasing function on [a,b].

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