8
G.W. JOHNSON and M.L. LAPIDUS
such spaces of measures, see, for example, [3, Chap. 4]. Given a Borel
subset B of (0,t), the total variation measure |n| is defined by
|n|(B) = sup
{^n_-i|
n(B.)I , where the supremum is taken over all finite
partitions of B by Borel sets (see [3, p. 126]). M(0,t) is a Banach
space under the natural operations and the norm
(0.8) ||n|| := |n|(0,t).
A measure y in M(0,t) is said to be continuous if y({x}) = 0 for every x
in (0,t). In contrast, v in M(0,t) is discrete (or is a "pure point
measure" in the terminology of Reed and Simon [42]) if and only if there
is an at most countable subset {x } of (0,t) and a summable sequence
{u } from C such that
(0.9) v = I a) 6 ,
P=l P P
where 6 is the Dirac measure with total mass one concentrated at x
T
P
p
[3, p. 12]. Every measure n e M(0,t) has a unique decomposition,
n = y - f v, into a continuous part y and a discrete part v [42, Theorem
1.13, p. 22]. We will make frequent use of such decompositions.
We work with the space M(0,t) throughout, but M[0,t] could be treated
without any essential complications. However, allowing n to have non-
zero mass at 0 introduces additional alternatives which we have chosen
to avoid.
G. L T Let n e M(0.t). A C-valued, Borel measurable function
°°1; n \ i / »
e on (0,t) x R is said to belong to L^-,
4
if
(0.10) HeiL* := / I I 0(s,.)IL d|n|(s) + -.
'»l;ri
(0,t)
oo
N
Note that if 6 e L
n
, then e(s,*) must be in L (K ) for n-a.e. s in
°°1; n
(0,t). If one makes the usual identification of functions which are
equal n x Lebesgue-a.e., the mixed norm space L^. , equipped with the
norm ll'll^i. » becomes a Banach space. Note that all bounded, every-
N
where defined, Borel measurable functions on (0,t) * E. are in L - ,
001;
n
for every n in M(0,t).
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