Canonical Banach function spaces generated by Urysohn universal spaces. Measures as Lipschitz maps

It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure CFL(U-r) of the linear span of the maps x bar right arrow d(x, a) - d(x, b), where d is the metric of the Urysohn space U-r of diameter r, is (isometrically if r = +infinity) isomorphic to the space LIP(U-r) of equivalence classes of all real-valued Lipschitz maps oil U-r. The space of all signed (real-valued) Borel measures oil U-r is isometrically embedded in the dual space of CFL(U-r) and it is shown that the image of the embedding is a proper weak* dense subspace of CFL(U-r)*. Some special properties of the space CFL(U-r) are established.