Real and Complex Analysis (Rudin)

COMPLEX MEASURES 117
This notation is perhaps not the best, but it is the customary one. Note that
I p.1 (E) ~ I p.(E) I, but that in general I p.1 (E) is not equal to I p.(E) I.
It turns out, as will be proved below, that I p.1 actually is a measure, so that
our problem does have a solution. The discussion which led to (3) shows then
clearly that I p.1 is the minimal solution, in the sense that any other solution A. has
the property A.(E) ~ I p.1 (E) for all E E IDl.
The set function I p.1 is called the total variation of p., or sometimes, to avoid
misunderstanding, the total variation measure. The term" total variation of p." is
also frequently used to denote the number I p.1 (X).
If p. is a positive measure, then of course I p.1 = p..
Besides being a measure, I p.1 has another unexpected property: I p.1 (X) < 00.
Since I p.(E) I :s; I p.1 (E) :s; I p.1 (X), this implies that every complex measure p. on
any cr-algebra is bounded: If the range of p. lies in the complex plane, then it
actually lies in some disc of finite radius. This property (proved in Theorem 6.4) is
sometimes expressed by saying that p. is of bounded variation.
6.2 Theorem The total variation I p.1 of a complex measure p. on IDl is a positive
measure on IDl.
PROOF Let {E;} be a partition of E E IDl. Let ti be real numbers such that
ti < I p.1 (EJ. Then each Ei has a partition {Aij} such that
L I p.(Aij) I > ti
)
(i = 1, 2, 3, ... ).
(1)
Since {Ai)} (i,j = 1, 2, 3, ... ) is a partition of E, it follows that
L ti :s; L I p.(Aij) I :s; I p.1 (E).
i i,)
(2)
Taking the supremum of the left side of (2), over all admissible choices of {t i},
we see that
L 1p.I(Ei):s; Ip. I (E). (3)
i
To prove the opposite inequality, let {A)} be any partition of E. Then for
any fixed j, {A j 11 E i } is a partition of A), and for any fixed i, {A) 11 Ei} is a
partition of E;. Hence
~ I p.(A)) I = ~I~ p.(Aj 11 E;)I
:s; L L 1p.(A) 11 EJI
) i
= L L 1p.(A) 11 E;)I:s; L 1p.I(Ei)·
; ) ;
(4)