2

NEIL E. GRETSKY

averaging condition, we have characterized B(M , X) , where M

is the closed subspace of L determined by the simple functions

of L . Here the representations are in terms of integrals of

scalar functions relative to additive vector-valued set functions

of p'-bounded variation (where p' is the norm dual to p) ,

using the integration theory of [2].

The problem of characterizing B(L ,2) in general appears

to be intractable by the present methods. However, if 3 6 is the

scalars, then B(L ,£) = L* admits a very detailed analysis and

is considered in Chapter III. There are two characterizations

given: one assumes the use of the averaging condition mentioned

above, the other does not. Both, however, use a further hypothesis

on the structure of the quotient space L /M • It is shown that

any continuous linear functional on L can be decomposed into

two parts, one of which annihilates M and the other of which

corresponds to a functional on M . (These are classically termed

singular and absolutely continuous, respectively.) The represent-

ations of these functionals are then obtained with methods which

generalize those used in Orlicz spaces. As a by-product a general

representation of functionals on M has been obtained with essen-

tially no restrictions on p .

2. Preliminaries. A brief outline of Banach function spaces

and some desired results will be presented here. Details of the

material below up to definition 13 may be found in [19, 20, 21,22].

Let (H,£,n) be a cr-finite measure space, and let M be

the collection of all non-negative measurable functions on Q

equipped with the usual pointwise (a.e.) order. (As usual, functions