Page 19 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
where a 1 b 1 a 2 b 2 ... a n b n i.e., a ’s and b ’s are forming finite collection
i
i
a ,b : i 1,2,...,n of pair-wise disjoint (non-overlapping) intervals (or of disjoint closed
i i
intervals).
Obviously, every absolutely continuous function is continuous.

Notes

 If a function satisfies f r b – f r a , even then it is absolutely continuous.

n
 The condition b a , means that total length of all the intervals must be
r r
r 1
less than .

2.1.2 Monotonic Function

Recall that a function f defined on an interval I is said to be monotonically non-increasing, iff

x y f x f y , x,y I

and monotonically non-decreasing, iff

x y f x f y , x,y I

Also f is said to be strictly decreasing, iff

x y f x f y

and strictly increasing, iff

x y f x f y

2.1.3 Functions of Bounded Variation – Definition

Let f be a real-valued function defined on [a,b] which is divided by means of points
a x x x ... x b.
0 1 2 n
Then the set P x ,x ,x ,...,x n is termed as subdivision or partition of [a,b].
1
2
0
b n 1 b b
Let us take V f,P f x f x , and V f,P supV f,P for all possible subdivisions P of
a r 1 r a a
r 0
b
[a,b]. (This is called total variation of f over [a,b] and also denoted by T f ).
a

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