0. Preliminarie s
Our terminolog y an d notatio n fo r Banac h space s an d linea r operator s i s standard; i t i s
similar t o tha t employe d b y Dunfor d an d Schwart z [D-SI] , an d Lindenstraus s an d Tzafrir i
[L-T2].
0.1. L p(p)-spaces and C{S)-spaces. Le t / i be a nonnegative (no t necessaril y sigm a fi-
nite) measur e define d o n a sigma field of subset s o f a set £2 . Le t 0 p » . B y L p((i)
(resp.
L P R(PL))
w e denote th e spac e of ju-equivalenc e classe s of complex-value d (resp . real-val-
ued) function s / suc h tha t \\f\\p °° , where
( Jftl/fd M ' f o r 0 p l ,
/ es s sup I/(5) | fo r p = °°.
{ s=S
For 1 p °° , Lp(ji) unde r th e nor m | ] || i s a Banac h space .
By C(S) (resp . CR(Sy* e denote th e Banac h spac e o f al l th e continuou s complex-val -
ued (resp . real-valued) function s o n a compact Hausdorf f spac e S wit h th e nor m ||/| | =
supje5|/(s)|. Give n a n / E C(S) w e define th e function s |/ | and/b y |/|(s ) = l/(s)|,/(s ) =
/(s)for s E S. Th e constan t function s ar e identified wit h th e scalars .
We identify th e dual spac e [C(5) ] * via the Ries z representatio n theore m wit h th e
space o f al l comple x Bore l measure s on S wit h th e nor m ||/i| | = th e tota l variatio n o f (JL. We
put [C(S)] % =
( M
G [C(S) ] *: ji t nonnegative}. A ^ [C(S)] X wit h
JLI(S )
=
||JU| |
- 1 is
called a probability measure. Give n a ju E [C(5) ] * an d a i E [C(S) ] * we write v « p i f *
is absolutely continuou s wit h respec t t o ju, and v 1 ji if v is singular wit h respec t t o ju. Le t
ju E [C(iS)]5j. . The n ther e i s a natural isometri c isomorphis m fro m L l(n) ont o th e subspac e
{v E. [C(S)] *: v « JJL] whic h assign s t o eac h g E LJ(ju) th e measur e i ; defined b y
(0.1) v(A) = I gdy. fo r ever y Bore l se t ACS.
J A
Giveni « /i , the uniqu e g- satisfying (0.1) is denoted b y dv\dy an d i t i s called th e Radon-
Nikodym derivative of v with respec t t o JJL. For a v E [COS) ] * we denot e b y \v\ the uniqu e
element o f [C(5) ] * suc h tha t v « \v\ and \dv(s)ld\v\\ = 1 M-almost everywher e on S. Fi -
nally give n a y E [C(S) ] * and a # E L^M) w e denot e b y g f th e uniqu e measur e i n
[CXS)] * whose Radon-Nikody m derivativ e wit h respec t t o \v\ \s g {dvld\v\)~
l.
3
http://dx.doi.org/10.1090/cbms/030/02
Previous Page Next Page