Page 166 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 13: Signed Measures

(iii) If <A > is any sequence of disjoint measurable sets, then Notes
n
 A n (A ),
n
n 1 n 1

i.e., is countably additive.
From this definition, it follows that a measure is a special case of a signed measure. Thus, every
measure on  is a signed measure but the converse is not true in general, i.e. every signed
measure is not a measure in general.
If – < (A) < , for very A , then we say that signed measure is finite.

13.1.2 Positive Set, Negative Set and Null Set

Definition
(a) Positive Set: Let (X, ) be a measurable space and let A be any subset of X. Then A X is
said to be a positive set relative to a signed measure defined on (X, ), if
(i) A , i.e. A is measurable.

(ii) (E) 0, E A s.t. E is measurable.
Obviously, it follows from the above definition that:
(i) every measurable subset of a positive set is a positive set,
(ii) is a positive set w.r.t. every signed measure.

Also for A to be positive. (A) 0 is the necessary condition, but not in general sufficient
for A to be positive.
(b) Negative Set: Let (X, ) be a measurable space. Then a subset A of X is said to be a negative
set relative to a signed measure defined on measurable space (X, ) if
(i) A  i.e., A is measurable.

(ii) (E) 0, E A s.t. E is measurable.
set A is negative w.r.t. , provided it is positive w.r.t. – .
(c) Null Set: A set A X is said to be a null set relative to a signed measure defined on
measurable space (X, ) is, A is both positive and negative relative to .
Thus, measure of every null set is zero.
Now, we know that a measurable set is a set of measure zero, iff every measurable subset of it
has measure zero. Thus, if A X is a null set relative to then (E) = 0, measurable subsets
E A. In other words.

A is a null set (E) = 0, measurable subsets E A.
Theorem 1: Countable union of positive sets w.r.t. a signed measure is positive.
Proof: Let (X, ) be a measurable space and let be a signed measure defined on (X, ). Let <A >
n
n
be a sequence of positive subsets of X, let A =  A i and let B be any measurable subset of A.
i 1
C
Set  = B A A C  A , n N.
n n n 1 1

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