Lower semicontinuity of monotone functionals in the mixed topology on $C_{b}$
证明了有界连续函数空间上单调泛函的从下方连续性等价于混合拓扑下的下半连续性，并由此得到凸单调泛函的可数可加测度对偶表示，对金融中的风险度量与超对冲问题有基础意义。
Abstract The main result of this paper characterises the continuity from below of monotone functionals on the space $C_{b}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> of bounded continuous functions on an arbitrary Polish space as lower semicontinuity in the mixed topology. In this particular situation, the mixed topology coincides with the Mackey topology for the dual pair $(C_{b},\mathrm{ca})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>ca</mml:mi> <mml:mo>)</mml:mo> </mml:math> , where $\mathrm{ca}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ca</mml:mi> </mml:math> denotes the space of all countably additive signed Borel measures of finite variation. Hence lower semicontinuity in the mixed topology is for convex monotone maps $C_{b}\to \mathbb{R}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>→</mml:mo> <mml:mi>R</mml:mi> </mml:math> equivalent to a dual representation in terms of countably additive measures. Such representations are of fundamental importance in finance, e.g. in the context of risk measures and superhedging problems. Based on the main result, regularity properties of capacities and dual representations of Choquet integrals in terms of countably additive measures for 2-alternating capacities are studied. Moreover, a well-known characterisation of star-shaped risk measures on $L^{\infty }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> is transferred to risk measures on $C_{b}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> . In a second step, the paper provides a characterisation of equicontinuity in the mixed topology for families of convex monotone maps. As a consequence, for every convex monotone map on $C_{b}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> taking values in a locally convex vector lattice, continuity in the mixed topology is equivalent to continuity on norm-bounded sets.