(p®x)(A) = p(Ax) in the field C of complex numbers. A similar remark applies
to the notion of weak-Cauchy norm-bounded nets. For X = y , B(X, Y) will be
abbreviated to B(X); and, if further X* = Y*, then LW(X, Y), BW(X, Y), and
BW(X, Y)* will be written as L^(X), ^ ( X ) and .B^pQ*, respectively. We now
make a few observations.
i.2.i. [3], p. 221, lines 15-20 say that if (F,Y*) satisfies 1.1.2, (iii) then
LW(X,Y) C B(X,Y) so that BW(X,Y) = LW(X,Y).
1.2.2. It follows from the first paragraph of the proof of [3], Proposition 1.6,
that if both (X,X*) and (Y,Y*) satisfy 1.1.2, (iii) (X* on ([3], p. 222, line 8) is
misprinted as Y*) then BW(X, Y) LW(X, Y) is closed in B(X, Y) and is, thus, a
Banach space. Thus, in this case, we may write the pair (BW(X, Y), BW(X, Y)*)
as (LW{X,Y), LW(X,Y)*), if we like.
1.2.3. The next few paragraphs of the proof of [3], Proposition 1.6, give that
if both (X, X*) and (Y,Y*) are compatible pairs then (LW(X,Y),LW(X,Y)*) is
an admissible pair satisfying 1.1.2, (iii). In particular, if (X, X*) is a compatible
pair then (LW(X),LW(X)*) is an admissible pair satisfying 1.1.2, (iii).
1.3. A compatible pair of spaces of functions and measures. Let S
be a locally compact Hausdorff space and M(S) be the Banach space of complex
regular Borel measures p on S of finite total variation \\p\\. It is in duality with
each of the subspaces Cb(S), Co(5) and CC(S) consisting of functions bounded,
zero at infinity, with compact support, respectively, of the linear space C(S) of
the continuous complex-valued function on 5. The C^S^-topology on M(5) is
usually referred to as the Bernoulli topology. The CC(S)-topology is the relative
vague topology. As in [4], Proposition 1.4, a net (pp)i3 in the subset M
consisting of non-negative measures in JA(S) converges to p G M(5) in the
Bernoulli topology if and only if (pp)p converges to p vaguely and furthermore,
Pp{S) converges to p(S). We denote the Bernoulli topology by rb.
1.3.1. It follows therefore that the Bernoulli topology rb coincides with the
cone topology [47], 2.2, on M+(Sr).
1.3.2. Furthermore, other results in [47], §2 suggest that it is convenient
to talk of the Bernoulli topology rb on M(S) rather than the cone topology on
(5) and r6-continuity rather than positive continuity. Moreover, it will be
handy in the use and further development of [3] in our set-up. Accordingly we
identify S as a closed subset of (M+(Sf), rb) via s p
, the (unit) point mass at
s. We note that the larger subset MP(S) of probability measures on S is also a
norm-bounded of norm one, r6-closed subset of M + (5).
1.3.3. The Strict Topology. Let /3 denote the strict topology on Cb(S)
introduced by R. C. Buck [11], the notation chosen because of its resemblance to
a topology used by A. Beurling [5]. J. B. Conway related f3 and compactness of
measures on S in [16], which is a detailed account of results announced in [15].
(a) By [11], Theorem 1, (Cb(S),/3) is complete, and /3-bounded and || Un -
bounded subsets of Cb(S) are the same. Moreover by [11], Theorem 2,
its dual is M(S).
(b) [16], Lemma 2.3, gives that the strong topology on M(S) is the norm
topology, i.e., the strong dual of (Cb{S),/3) and the dual of (Co(5), ||-||Sup)
Previous Page Next Page