A sufficient condition for a singular functional on L-infinity [0,1] to be represented on C [0,1] by a singular measure

作者：Wrobel Andrew J^*

来源：Indagationes Mathematicae, 2018, 29(2): 746-751.

DOI：10.1016/j.indag.2017.12.005

• Singular functional
• Singular measure
• Yosida-Hewitt decomposition
• Lebesgue decomposition

As is known, there are singular linear functionals on L-infinity [0, 1] whose restrictions to C [0, 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 C [0, 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.

• 出版日期2018-4