REPRESENTATION THEOREMS ON BANACH FUNCTION SPACES 7
A partition £ is defined to be a finite disjoint collection of
non ia-null members of £ which are of finite measure. Define
o
the "averaged" step function of f L to be
f
e=KI |f|Wn(E.)E-
Ei
DEFINITION 13. A function norm is said to have property (J)
if, for each partition £ , p(f£) £ p(f)
LEMMA 14. If the collection of partitions is regarded as
partially ordered by refinement, and if p has (J),then p(f£)
is an increasing function of £ .
Proof. Let £= {E.|i=l,...,n} and 3 = {F.|j=l,...,m] be two
partitions such that £ is finer than 3 ; i.e. each E. is either
contained in some F. or disjoint from all F. , and each F. is
J
J
J J
the union of a collection of the E. . Renumbering, if necessary,
we define k ,...,km such that k = 0; E,. E, c F. and
o o ' k
n+.i
1 k . i
k.
U E. = F. for j=l,...,m ; and E, ,
n
,...,E are disjoint
i-kj.j+l x J V" 1 n
from all F. . A simple computation yields that
(fp)g(=
f* *
anc*
thus pCf^) = p((fe)^) *
p fe by (J)- QED
REMARK 15. i) Property (J) is satisfied by all the well
known Banach function spaces, e.g. all the Orlicz spaces (and in
particular the Lebesgue spaces).
ii) Property (J) is similar to a property, called levelling,
in [7];it was remarked in [7] that if a function norm is levelling
then it satisfies (J). No use will be made of this remark below.
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