Page 171 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes  Let (X, ) be a measurable space and then A X is said to be a positive set relative to a
signed measure defined on (X, ) if
(i) A is measurable

(ii) (E) 0, E A s.t. E is measurable.
 Let (X, ) be a measurable space. Then A X is said to be negative set relative to a signed
measure if
(i) A is measurable
(ii) (E) 0, E A s.t. E is measurable.

 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 .

13.3 Keywords

Hahn Decomposition: Definition: A decomposition of a measurable space X into two subsets s.t.
X = P Q, P Q = .
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.

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 .
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.
Signed Measure: Let the couple (X, ) be a measurable space, where  represents a -algebra of
subsets of X. An extended real valued set function

:  [– , ]
defined on  is called a signed measure, if it satisfies the following postulates:
(i) assumes at most one of the values – or + .

(ii) ( ) = 0.
13.4 Review Questions

2
x
1. If (E) = xe dx, then find positive, negative and null sets w.r.t. . Also give a Hahn
E
decomposition of R w.r.t. .
2. State and prove Hahn decomposition theorem for signed measures.
3. If is a measure and , are the signed measures given by (E) = (A E), (E) = (B
1 2 1 2
E), where (A B) = 0, show that .
1 2

164 LOVELY PROFESSIONAL UNIVERSITY