Duality in Measure Theory

Let X be a set, let R be a 6-ring* of subsets of X, let M be the vector lattice of real measures on R , and let M ~ be the = dual of M, i.e. the vector lattice of order continuous linear forms on M. In order to see why the elements of M ~ are considered as a kind of generalized functions on X, let us denote by F the vector lattice of real functions on x which are universally integrable with respect to any measure of M, i.e.
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