REPRESENTATION THEOREMS ON BANACH FUNCTION SPACES 3

differing only on (a-null sets will be identified so that the

elements of M are in reality equivalence classes of functions.)

DEFINITION 1. A mapping p on M to the extended reals

is called a function norm if p satisfies the following conditions

for all f and g in M :

i) p(f) ; 0 and p(f) » 0 if and only if f = 0

ii) p(otf) = ocp(f) for a ; 0

iii) p(f + g) ^ p(f) + p(g)

iv) f £ g in M implies p(f) p(g) .

The definition of p is extended to M , the collection of

all complex valued functions on Q , by defining p(f) = p(|f|)

for f € M . In order to avoid pathological cases, only non-

trivial p will be considered, i.e. we require that there exists

f e M such that 0 p(f) ° ° .

DEFINITION 2. i) A function norm p has the weak Fatou

property (WFP) if f tf and sup P(fn) ° ° imply p(f) «

where f € M+ and f € M+, n = 1,2,... .

n

ii) A function norm p has the strong Fatou property

(SFP) if f tf implies p(fn)tp(f) where f €

M+

and fn €

M+,

n « 1,2,... . Note that SFP implies WFP, but the converse need

not be true.

DEFINITION 3. Let Lp(Q,Z,[i) = {f € M| p(f) « } .

It is clear that L is a normed linear space and, in fact,