Random Linear Functionals

R. M. Dudley
1969 Transactions of the American Mathematical Society  
Let S be a real linear space (=vector space). Let Sa be the linear space of all linear functionals on S (not just continuous ones even if S is topological). Let 86{Sa, S) or @{S) denote the smallest a-algebra of subsets of Sa such that the evaluation x -> x(s) is measurable for each s e S. We define a random linear functional (r.l.f.) over S as a probability measure P on &(S), or more formally as the pair (Sa, P). In the cases we consider, S is generally infinite-dimensional, and in some ways
more » ... is too large for convenient handling. Various alternate characterizations of r.l.f.'s are useful and will be discussed in §1 below. If 7 is a topological linear space let 7" be the topological dual space of all continuous linear functions on T. If 7" = S, then a random linear functional over S defines a " weak distribution " on 7 in the sense of I. E. Segal [23] . See the discussion of "semimeasures" in §1 below for the relevant construction. A central question about an r.l.f. (Sa, P), supposing S is a topological linear space, is whether P gives outer measure 1 to S'. Then we call the r.l.f. canonical, and as is well known P can be restricted to S' suitably (cf. 2.3 below). For simplicity, suppose that for each s e S, j x(s)2 dP(x) <oo. Let C(s)(x) = x(s). Then C is a linear operator from S into the Hilbert space L\Sa, P). One seeks conditions on this operator to make the r.l.f. canonical. If S itself is a Hilbert space, the fundamental theorem of Sazonov [22] and associated results due to Minlos [20] and others assert that it is sufficient for C to be a Hilbert-Schmidt operator, and that this result is the best possible. If S is a Banach space, it is sufficient for C to be nuclear in the sense of Grothendieck [15]; see 8.5 below. The conjecture that it would suffice for C*C to be nuclear, as if S is Hilbert, is not true even in the Gaussian (mean 0) case, where J" eix{s) dP(x) = e~Q(-s), Q being a nonnegative definite quadratic form on the Banach space S; see the end of §6 below. In another paper [8], I have, in effect, studied conditions for a Gaussian r.l.f. over a Banach space to be canonical. It appears that the relevant conditions concern the "size" of C(M), where M is the unit ball of S. Here the size may be measured by £-entropy or by volumes of finite-dimensional projections. Conditions like nuclearity of C seem poorly adapted to give precise results here since they measure the size of C(M) only in certain rather arbitrary directions.
doi:10.2307/1994697 fatcat:iwvjbntwxzhepjxmcf6tcoxtya