Introduction. By space of continuous functioins we mean the Banach space C of real continuous functions on a compact Hausdorff space X. C, its first dual, and its second dual are not only Banach spaces, but also vector lattices, and this aspect of them plays a central role in our treatment. In ??1-3, we collect the properties of vector lattices which we need in the paper. In ?4 we add some properties of the first dual of C-the space of Radon measures on X-denoted by L in the present paper. In ?5 we imbed C in its second dual, denoted by M, and then identify the space of all bounded real functions on X with a quotient space of M, in fact with a topological direct summand. Thus each bounded function on X represents an entire class of elements of M. The value of an element f of M on an element ,i of L is denoted as usual by f(,u). If f lies in C then of course f(,u) =,u(f) (usually written f fd,u), and thus the multiplication between M and L is an of the multiplication between L and C. Now in integration theory, for each ,uEL, there is a standard extension of ffd,u to certain of the bounded functions on X (we confine ourselves to bounded functions in this paper), which are consequently called ,i-integrable. The question arises: how is this extension related to the multiplication between M and L. ??6-8 (and ?12) are devoted to this question. We show that given a bounded function f on X the equivalence class of M represented by f contains two distinguished elements, which we denote by f* and f*, such that for every Radon measure ,u, f*(,u) = f*fd,u and f*(,u) = f*fdM, where f* and f* are the lower and upper integrals of f with respect to ,u. Moreover, if f is integrable for every Radon measure, then f* and f* coincide, and we have a unique element whose value on each ,u is ffd4. In ?9 we study order-convergence in M. It is in this area that Af offers its most striking immediate gain in clarity over consideration of only the bounded functions oil X. We cite two examples. The first example: The characteristic functions of finite subsets of the real initerval 0< x < 1 form a directed system which order-converges to the constant function 1, while their Lebesgue integrals, being all zero, do not converge to the Lebesgue integral of 1. In M, however, this directed system order-converges to an element different from 1, and the above unsatisfactory situation disappears. In point of fact, order-convergence is perfectly well-behaved in M: if a directed system order converges, its values on every Radon measure converge