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AJIT IQBAL SINGH

(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

+

(5)

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

M

+

(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

s

, 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)