Page 9 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes 1.1 Differentiation and Integration

1.1.1 Lipschitz Condition

Definition: A function f defined on [a, b] is said to satisfy Lipschitz condition (or Lipschitzian
function), if a constant M > 0 s.t.

|f (x) – f (y)| M |x – y|, x, y [a, b].

1.1.2 Lebesgue Point of a Function

Definition: A point x is said to be a Lebesgue point of the function f (t), if

1 x n
Lim |f(t) f(x)|dt 0.
h 0 h x
1.1.3 Covering in the Sense of Vitali

Definition: A set E is said to be covered in the sense of Vitali by a family of intervals (may be open,
closed or half open), M in which none is a singleton set, if every point of the set E is contained in
some small interval of M i.e., for each x E, and > 0, an interval I M s.t. x I and (I) .

The family M is called the Vitali Cover of set E.

Example: If E = {q : q is a rational number in the interval [a, b]}, then the family I
q i
1 1
where I q , q , i N is a vitali cover of [a, b].
q i
i i
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,
k
… n} of M, such that
n

m * E I k < .
k 1
Proof: Without any loss of generality, we assume that every interval of family M is a closed
interval, because if not we replace each interval by its closure and observe that the set of end
points of I , I , …… I has measure zero.
1 2 n
[Due to this property some authors take family M of closed intervals in the definition of Vitali’s
covering].
Suppose 0 is an open set containing E s.t. m* (0) < m* (E) + 1 < we assume that each interval in
M is contained in 0, if this can be achieved by discarding the intervals of M extending beyond 0
and still the family M will cover the set E in the sense of Vitali.
Now we shall use the induction method to determine the sequence <I : k = 1, 2, … n> of disjoint
k
intervals of M as follows:

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