Real and Complex Analysis (Rudin)

COMPLEX MEASURES 119
We have thus split E into disjoint sets A and B with I J.L(A) I > 1 and
I J.L{B) I > 1. Evidently, at least one of I J.L I (A) and I J.L I (B) is 00, by Theorem 6.2.
Now if IJ.LI(X) = 00, split X into A 1, Blo as above, with I J.L(A 1 ) I > 1,
I J.L I (B 1) = 00. Split B 1 into A 2 , B 2 , with I J.L(A 2)1 > 1, I J.L I(B 2 ) = 00. Continuing
in this way, we get a countably infinite disjoint collection {Ai}, with
I J.L(Ai) I > 1 for each i. The countable additivity of J.L implies that
But this series cannot converge, since J.L(Ai) does not tend to 0 as i-+ 00. This
contradiction shows that I J.L I (X) < 00.
IIII
6.S If J.L and A are complex measures on the same a-algebra IDl, we define J.L + A
and CJ.L by
(J.L + AXE) = J.L(E) + A(E)
(CJ.L)(E) = cJ.L(E)
(E E IDl) (1)
for any scalar c, in the usual manner. It is then trivial to verify that J.L + A and CJ.L
are complex measures. The collection of all complex measures on IDl is thus a
vector space. If we put
IIJ.LII = I J.L I (X), (2)
it is easy to verify that all axioms of a normed linear space are satisfied.
6.6 Positive and Negative Variations Let us now specialize and consider a real
measure J.L on a a-algebra IDl. (Such measures are frequently called signed measures.)
Define I J.L I as before, and define
Then both J.L + and J.L - are positive measures on IDl, and they are bounded, by
Theorem 6.4. Also,
The measures J.L + and J.L - are called the positive and negative variations of J.L,
respectively. This representation of J.L as the difference of the positive measures J.L +
and J.L- is known as the Jordan decomposition of J.L. Among all representations of
J.L as a difference of two positive measures, the Jordan decomposition has a
certain minimum property which will be established as a corollary to Theorem
6.14.
(1)
(2)