6 NEIL E. GRETSKY
vii) L / is isometrically isomorphic to a closed sub-
space of L* (the conjugate space of LQ) since if g L /
then G(f) = JQ fgd|i defines G £ L* such that ||G| | = p'(g) .
This subspace is not in general all of L* . The correspondence
is also a lattice isomorphism.
DEFINITION 9. An element f 6 L has absolutely continuous
norm if
a) lim P(xE ) = 0 for any sequence {E } such that E c E
n
with p(xE) ° ° and lim [i(E ) = 0 , and
b) lim P(fxQ-Q ) - 0 for any admissible sequence {Q } .
n
Let L^ = {f L If has absolutely continuous norm} . Note
that IP is a closed linear subspace and a normal sublattice of L„
P * P
DEFINITION 10. Let T T be an admissible sequence {Q } .
Define Lj! = "sp {f L If is bounded and has support in some 0 } .
Note that for each T T , \T is a closed linear subspace and
a sublattice of L
P
REMARKS 11. i ) It can be shown that n LV = L^
ii) Although L^ is not necessarily norm determining,
each L^ is. (A subspace Y c L is norm determining if
p'(g) = sup {J|fg|dn | p(f) 1, f e Y} for all g Lp, .)
iii) (LQ)* L / if and only if there is an admissible
sequence T T such that L™ = L„ .
H
P P
DEFINITION 12. Define SQ = {E S|p(x
E
) °°}
S^ = {E s | p ' ( x
E
) °°}
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