Real and Complex Analysis (Rudin)

154 REAL AND COMPLEX ANALYSIS
(ii) X is Lebesgue measurable, T is one-to-one on X, and T is differentiable at
every point of X;
(iii) m(T(V - X» = o.
Then, setting Y = T(X),
for every measurable f: Rk --+ [0, 00].
If dm = i Uo T)IJTI dm (1)
The case X = V is perhaps the most interesting one. As regards condition
(iii), it holds, for instance, when m(V - X) = 0 and T satisfies the hypotheses of
Lemma 7.25 on V-X.
The proof has some elements in common with that of the implication
(b)--+ (c) in Theorem 7.18.
It will be important in this proof to distinguish between Borel sets and
Lebesgue measurable sets. The u-algebra consisting of the Lebesgue measurable
subsets of Rk will be denoted by rot
PROOF We break the proof into the following three steps:
(I) If E E 9Jl and E c V, then T(E) E 9Jl.
(II) For every E E 9Jl,
(III) For every A E 9Jl,
m(T(E 11 X» = iXEIJTI dm.
1 XA dm = i (XA 0
T) I J T I dm.
If Eo E 9Jl, Eo c V, and m(Eo) = 0, then m(T(Eo - X» = 0 by (iii), and
m(T(Eo 11 X» = 0 by Lemma 7.25. Thus m(T(Eo» = O.
If E 1 C V is an F", then E 1 is u-compact, hence T(E 1) is u-compact,
because T is continuous. Thus T(E 1 ) E 9Jl.
Since every E E 9Jl is the union of an F" and a set of measure 0
(Theorem 2.20), (I) is proved.
To prove (II), let n be a positive integer, and put
Because of (I), we can define
v,,={XE V: I T(x) I