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).
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