Page 170 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 13: Signed Measures

Now, Q – E is a subset of Q, it follows that Notes
n
(Q – E ) 0.
n
Since (Q – E ) E =
n n
and Q = (Q – E ) E ,
n n
we have (Q) = (Q – E ) + (E )
n n
(Q) (E ), n N and E B.
n n
Therefore (Q) K. … (iii)

(ii) and (iii) (Q) = K – < k. … (iv)
C
Now we shall show that p = Q , the complement of Q w.r.t. is a positive subset of X. Suppose
not, i.e. P is negative. Then E P s.t. E is measurable and (E) < 0. Now we know that if
– < (E) < 0, we get a negative set A E s.t. (A) < 0.
A, Q are distinct negative subsets of X

A Q is negative set
(A Q) K [using (i)]
(A) + (Q) K,
(A) + K K, [using (iv)]

(A) 0,
a contradiction, for (A) < 0
P = Q is a positive subset of X
C
Q is a negative subset of X.

Thus X = P Q, P Q = .

13.1.4 Hahn Decomposition: Definition

A decomposition of a measurable space X into two subsets s.t. X = P Q, P Q = ,
where P and Q are positive and negative sets respectively relative to the signal measure , is
called as Hahn decomposition for the signed measure . P and Q are respectively called positive
and negative components of X.
Notice that Hahn decomposition is not unique.

13.2 Summary

 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.
(iii) If <A > is any sequence of disjoint measurable sets, then is countably additive.
n

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