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Introduction. By "the space of continuous functions" 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 presentdoi:10.2307/1992866 fatcat:e56j7txavbfudkehl27ppu4jsq