Real and Complex Analysis (Rudin)

126 REAL AND COMPLEX ANALYSIS
A u B = X, A ("'\ B = 0, and such that the positive and negative variations
J1. + and J1. - of J1. satisfy
(E E rol). (1)
In other words, X is the union of two disjoint measurable sets A and B, such
that "A carries all the positive mass of J1." [since (1) implies that J1.(E) ~ 0 if
E c A] and" B carries all the negative mass of J1." [since J1.(E) :s; 0 if E c B]. The
pair (A, B) is called a Hahn decomposition of X, induced by J1..
PROOF By Theorem 6.12, dJ1. = h d I J1.1, where I h I = 1. Since J1. is real, it follows
that h is real (a.e., and therefore everywhere, by redefining on a set of
measure 0), hence h = ± 1. Put
A = {x: h(x) = I},
Since J1. + = 1< I J1.1 + J1.), and since
we have, for any E E rol,
1