Real and Complex Analysis (Rudin)

132 REAL AND COMPLEX ANALYSIS
There are complex numbers IX;, I IX; I = 1, so that lXi (hJ = I (h;) I, i = 1, 2.
Then
AI + Ag :::; I (hl) I + I (h2) I + 2e
= (1X1 hI + 1X2 h2) + 2e
:::; A( I hI I + I h 2 1) + 2e
:::; A(I + g) + 2e,
so that the inequality ~ holds in (10).
Next, choose h E Cc(X), subject only to the condition
V = {x:/(x) + g(x) > O}, and define
h (x) = I(x)h(x) , h 2 (x) = g(x)h(x)
1 I(x) + g(x) I(x) + g(x)
h 1 (x) = hix) = 0
(x ¢ V).
I h 1:::;1 + g, let
(x E V),
It is clear that hI is continuous at every point of V. If Xo ¢ V, then h(xo) = 0;
since h is continuous and since I hl(X) I :::; I h(x) I for all x E X, it follows that
Xo is a point of continuity of hI' Thus hI E Cc(X), and the same holds for h2 .
Since hI + h2 = h and I hI I :::;f, I h21 :::; g, we have
I (h) I = I (hl) + (h2) I :::; I (hl) I + I (h2) I :::; AI + Ag.
Hence A(I + g) :::; AI + Ag, and we have proved (10).
If I is now a real function, IE Cc(X), then 21 + = I I I + f, so that 1+ E
Cc+(X); likewise,f- E Cc+(X); and since I =1 + - 1-, it is natural to define
and
(12)
(I E Cc(X),f real) (13)
A(u + iv) = Au + iAv. (14)
Simple algebraic manipulations, just like those which occur in the proof of
Theorem 1.32, show now that our extended functional A is linear on Cc(X).
This completes the proof.
IIII
Exercises
I If Il is a complex measure on a a-algebra !lJl, and if E E !lJl, define
J.(E) = sup L I Il(E I) I,
the supremum being taken over all finite partitions {E 1} of E. Does it follow that J. = I III ?
2 Prove that the example given at the end of Sec. 6.10 has the stated properties.
3 Prove that the vector space M(X) of all complex regular Borel measures on a locally compact
Hausdorff space X is a Banach space if IIllil = III I (X). Hint: Compare Exercise 8, Chap. 5. [That the
difference of any two members of M(X) is in M(X) was used in the first paragraph of the proof of
Theorem 6.19; supply a proof of this fact.]