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).