A sufficient condition for a singular functional on L∞0,1 to be represented on C0,1 by a singular measure

As is known, there are singular linear functionals on L∞0,1 whose restrictions to C0,1 are represented by densities. But it is shown here that a singular functional cannot be of this “diffuse” kind if an evanescent sequence of sets that support it can be chosen to consist of closed (rather than merely measurable) sets: the restriction to C0,1 is then represented by a measure singular with respect to the Lebesgue measure. This can furnish a more tractable representation of singular Lagrange multipliers and hence, in economic theory, of extremely concentrated capital charges. The results remain true when 0,1 is replaced by any compact,T, with an “underlying” finite nonatomic Borel measure.