8 NEIL E. GRETSKY
There are some properties of function norms with property
(J) that are needed and which are proved in [7] under the levelling
hypothesis. Since the arguments arevirtually the same in the two
contexts they will only be sketched here.
LEMMA 16. If p ha£ (J), then
a) P(xE) * ^EUF) p(xF) for, E,F of finite measure,
b) if _ 0 ia(E) « then 0 P(xE) ° ° ,
c) if [i(E) - then |a(E)= p(x£)
P'(X£)
.
Proof, a) If f = XTT and if £ is the one set partition
{E U F} , where E and F are of finite measure, then
p(xE * p(^EUFj XEUF)= ]I^EUFT
P ^EUF )
*
M^EOFT
p(xF *
b) Since p is a norm on equivalence classes of functions differ-
ing on |j.-nul l sets, M-(E)=0 if and only if
p(xp)=
0 Assume
|a(E) » . Then taking F such that 0 ia(F) » and p(xp) °°
we have p(xE) * j^/F)
P(F)
° °
hY Part a-
c) If |i(E)= 0 , the statement is trivial. Let 0 |i(E) .
n
If g is a simple function in L , say g = £ a.Xp
anc*
^
e
P i=i
1

n
i s the one set partitio n [E f] U E.} , then P |gp|d^i = P |g|d(j .
i=l
x
E
6
E
Moreover p(g£) £ p(g) by (J). Thus,
p'(xE) = sup {J |ge|d|j j p(g) £ 1 and g simple}
= sup {J kdjj I p(kx£) £ 1}
= i^EMx^"1 . QED.
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