4 NEIL E. GRETSKY

is a (complex) AB-lattice. In general, however, it is not complete

without strengthening the hypotheses on p . Examples may be

found in [20]. A sufficient condition turns out to be the weak

Fatou property.

In what follows, it is always assumed without further mention

that p has the weak Fatou property.

DEFINITION 4. A sequence of sets [f t } such that Q € E,

ft to, |a(Q ) °°, and P(xQ ) ° ° is called admissible with respect

n

to p .

In order to guarantee the existence of admissible sequences

the underlying measure space may have to be adjusted as follows.

DEFINITION 5. A set E c Q is called unfriendly (relative

to p) if for every measurable B c E , |^(B) 0 implies p(xR) = ° ° •

It can be shown that there is a measurable E c ft which is

maximal with respect to the property of being unfriendly. This

set is removed from ft and the new a-finite measure space will

be relabeled as (ft,E,[i) . Note that the removal of E does not

really restrict the generality since, if f € L and E is un-

friendly, then f(ua) = 0 for almost all u ) £ E . In the "new"

(ft,H,|j) there are always admissible sequences.

THEOREM 6. Let L be a function space on a a-finite measure

space (n,S,|a) with no unfriendly sets as above. Then

a) There is a constant y , 0 Y ^ 1 , such that if {fn} is

any sequence in M , f 6 M , and f tf , then YP(f) ^ l^m

P(fn)

;

b) L is complete (and is called a Banach function space).