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

) °°} •