Closure properties of 𝐶(𝑋) in its second dual

Abstract

1. Let X be a fixed compact Hausdorff space, C the Banach lattice of continuous real functions on X, L its dual, and M its second dual. The closure of C in M under the weak topology w(M, L) determined by L is, as is well known, M itself. In fact, the closure of C under the Mackey topology r(M, L) is M itself, by the Grothendieck Theorem. A deeper property is that this relationship between C and M also holds under order-convergence, which is finer than r(M, L)-convergence. Let us examine this in more detail. An order-bounded net {f, } in M converges to fEM if f = lim infa fa = lim supa fa, where lim infa fa = Va (A#>a f,) and lim supa fa = A a(V#>a f,). If a set A contains all such limits of order-convergent nets of A, we say A is closed under order-convergence, or simply closed. In general a set A is not closed. However there exists a smallest closed set containing A, and it is this set which is called the closuire of A. With this definition we have the property stated above: the closure of C under order-convergence is M itself. Unlike the case of topological convergence, the closure of a set A cannot in general be obtained by adjoining to A all limits of orderconvergent nets of A. If we adjoin all such limits, the enlarged set need not be closed, and it may be necessary to iterate the process repeatedly, possibly a transfinite number of times, before every point of the closure is obtained. In particular, the set obtained by adjoining to C all limits of order-convergent nets of C is a proper subset of M. In [2 ] we gave this set the symbol U, since it corresponds to those bounded functions on X which are integrable with respect to every Radon measure (universally integrable), and we have studied its properties in that and subsequent papers. Beyond U we did not go, only conjecturing [3, ?4] that it would require an uncountable number of iterations of the above process to obtain all of M. Contrary to the conjecture, we now present a proof that every element of M is the limit of an order-convergent net of U. Thus all of M can be obtained from C by one iteration of the operation of adjoining limits of order-convergent nets.