Page 55 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes 4.2 Summary

 A real-valued function f defined on [a,b] is said to be absolutely continuous on [a,b], if for
an arbitrary 0, however small, a, 0, such that

n n
f b f a , whenever b a .
r r r r
r 1 r 1
 Every absolutely continuous function is continuous.
 Every absolutely continuous function f defined on [a,b] is of bounded variation.

4.3 Keywords

Absolute Continuity of Functions: Absolute continuity of functions is a smoothness property
which is stricter than continuity and uniform continuity.
Absolute Continuous Function: A real-valued function f defined on [a,b] is said to be absolutely
continuous on [a,b], if for an arbitrary 0, however small, a, 0, such that

n n
f b f a b a ,
r r whenever r r
r 1 r 1
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 intervals.
i i
4.4 Review Questions

1. Define absolute continuity for a real variable. Show that f(x) is an indefinite integral, if F
is absolutely continuous.

2. If f,g: [0,1] R are absolutely continuous, prove that f + g and fg are also absolutely
continuous.

3. Show that the set of all absolutely continuous functions on an interval I is a linear space.
4. If g is a non-decreasing absolutely continuous function on [a,b] and f is absolutely continuous
on [g(a), g(b)], show that fog is also absolutely continuous on [a,b].
5. If f is absolutely continuous on [a,b] and f' x 0 for almost all x [a,b], show that f is
non-decreasing on [a,b].

4.5 Further Readings

Books Krishna B Athreya, N Soumendra Lahiri, Measure Theory and Probability Theory,
Springer (2006).

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