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.