REPRESENTATION THEOREMS ON BANACH FUNCTION SPACES 9

COROLLARY 17. If. P M l O) then EQ n % = £^ n Sp , where

Sp = {E € E||i(E) °° } . In particular, every partition hasall

its members in both £ and £' .

o o

COROLLARY 18. If. P has^ (J) and_ u(Q)= • , then |a(E) = •

implies that p(xF)= P(x0) (which may be finite or infinite).

Proof. There exists an increasing sequence {E } such that

li(En) » and l^m H(E ) = |a(E) = ° ° . Take {Qm3 to be a p-admis-

sible sequence. Then

H(E_)

P(XE) * P(XE ) *

(Q UE

) P(XQ ) for m,n=l,2,...

n N m n' m

by lemma 16 (a). Thus

P*E * SUP L U(E UQ ) p(*Qm)J = PV •

m,n v n m

The reverse inequality is always true. QED.

LEMMA 19. I| _ P has property (J),then p' has property(J).

Proof. Let f € L„ and h 6 L„/ . Then

P P

J|fh£|d|i = J|f| YQ Ihldn/nCE^Xj.. d^

e E i i

= Y ( L |h|dn) (J" |f|d

M

)/

M

(E.)

e i i

- J X ( f |h|dM/U(Ei))(f Ifldn/nCE^j. dn

e

E

i

E

i *

- J I K X

ihid^EiXJ[ZCf

ifMM/^(E.))xE.

e E i x e E i

= J|fehe|dM .

Similarly, J|f

e

h|du = J|f

£

h

e

|d|i , and the*fore J * | f

£

ri | d^t = $\fti&\6\i