As d e f in e d by ( 9 . 1 ) , weak c o n v e rg e n c e can be i n t e r p r e t e d as c o n v e rg e n c e in th e weak* to p o lo g y on t h e d u a l o f t h e Banach s p a c e C(K) , s i n c e B o re l p r o b a b i l i t i e s can be i d e n t i f i e d w ith c e r t a i n l i n e a r f u n c t i o n a l s on C(K)
( o fo th e comments o f B i l l i n g s l e y (1 9 6 8 , p . 1 6 ) ) . T h is i d e a can be c a r r i e d o v e r t o g e n e r a l t o p o l o g i c a l s p a c e s , t o g iv e a q u i t e c o m p re h en siv e th e o r y f o r weak c o n v e rg e n c e o f B a ir e m e a s u re s . Such a tr e a t m e n t was g iv e n by V a r a d a r a ja n ( 1 9 6 5 ) , b a s e d on a c o m b in a tio n o f th e g e o m e tr ic a p p ro a c h o f A le x a n d r o f f (1 9 4 0 , 1941, 1943) and th e f u n c t i o n a l a n a l y t i c m ethods o f Le Cam
(1957). It is essentially this form of the theory which we shall be describing, but our efforts will be channelled towards making greater use of simple functional analytic ideas. When combined with the integral representation techniques of Chapter 2, these lead to a variety of results in measure theoretic form, results which are difficult to obtain by more direct arguments.
We work from the general to the particular, starting with an abstract formulation for which the functional analytic aspects are to the fore. To begin with we consider the totality
L
of positive linear functionals on acertain space C of real valued functions,
L
being equipped with the natural weak* type 0 f topology. Certain subspacesL(V) of L
will be shown to correspond to particular types of measures, such as the G-additive or tight ones. Identification of the relatively compact subsets ofL(V)
can be reduced to two simple tasks: show that
A
czL(V
) is relatively compact inL
, and then that the closure ofA
in L contains only members ofLCD)
. The results can then be translated into measuretheoretic terms by means of the theorems of Chapter 2 U In spirit, but not in actual manipulative detail, this approach is close to that adopted by Le Cam (1957), and later Bourbaki (1969, p. 63).
Suppose then that we are given a vector lattice of real valued functions on some abstract space X . Let C denote its positive cone. We assume that' C also satisfies Stone’s condition (see A3 of Section 3), in order that a good proportion of the conditions of Theorem 5.1 should be satisfied. This will ensure that there is a rich supply of measures
generated by the cone
L of
all non-negative linear functionals on C . Possible choices for C include the bounded continuous functions on an arbitrary topological space (as in Varadarajan’s (1965) paper), thecontinuous functions of compact support on a locally compact space (leading to the concept of vague convergence), or the bounded uniformly continuous functions on some uniform space (see Chapter 4).
Equip
L
with itsC-topology
i.e, the weakest topology such thatTig)
is a continuous function ofT
€L
, for everyg
€ C . A net {T^} converges toT
in this topology iff 21(g0 -*■Tig)
for everyg
€ C ; so the idea of weak convergence (or more properly, weak* convergence) is underlying the definition. We shall sometimes use the prefix C- when describing properties of this topology e.g. C-relatively compact means relatively compact wrt the C-topology.Now to identify those members of L which correspond to measures of varying degrees of smoothness, we make use of a device similar to that employed in Examples 6.3 and 6.4. Take the case of tight Borel measures for example. Such measures can usually be identified with those positive linear functionals on C ( X ) for which: ® for anY net in
C ( X ) such that 1 > -*■ 0 uniformly on compacta. Notice that the map
T I— *■ is continuous in T for each fixed f^ . This generalises immediately to the following procedure.
Suppose V is a family of nets of real valued C-continuous functions