Page 16 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 1: Differentiation and Integration: Differentiation of Monotone Functions

Notes
1 x o h
or Lim |f (t) f(x )|dt 0 [ Modulus of any quantity is always non-negative]
o
h o h
x o
1 x o h
i.e. Lim |f (t) f(x )|dt 0 .
o
h o h
x o
This shows that x is a Lebesgue point of f (t).
o
1.2 Summary

 A function f defined on [a, b] is said to satisfy Lipschitz condition if a constant M > 0 such
that

| f (x) – f (y) M |x – y|, x, y [a, b].
 A point x is said to be a Lebesgue point of the function f (t), if
1 x h
Lim f(t) f(x) dt 0 = 0
h o h x
 Let E be a set of finite outer measure and M be a family of intervals which cover E in the
sense of Vitali; then for a given > 0, it is possible to find a finite family of disjoint
intervals
{I , k = 1, 2, …, n} of M, such that
k
n

m * E I k .
k 1
 Lebesgue differentiation theorem: Let f : [a, b] R be a finite valued monotonically
increasing function, then f is differentiable. Also
f : [a, b] R is L-integrable and

b
f (x) dx f(b) f(a) .
a
1.3 Keywords

Dinni’s Derivatives: These are the ways to define the quantities to judge the
measurability of the functions even at the points where it is not differentiable.
Fundamental Theorem of the Integral: The fundamental theorem of the integral calculus is that
differentiation and integration are inverse processes.

Measurable functions: An extended real valued function f defined over a measurable set E is said
to be measurable in the sense of Lebesgue if set
E (f > a) = {x E : f (x) > a} is measurable for all extended real numbers a.

Vitali's Lemma: Let E be a set of finite outer measure and M be a family of intervals which cover
E in the sense of Vitali; then for a given > 0, it is possible to find a finite family of disjoint
intervals {I , k = 1, 2, … n} of M, such that
k
n

m * E I k < .
k 1

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