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
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