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
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