Quantitative approximation by nonlinear convolution operators of Landau-Choquet type

By using the concept of Choquet nonlinear integral with respect to a monotone set function, we introduce the nonlinear convolution operators of Landau-Choquet type, with respect to a family of submodular set functions. Quantitative approximation results in terms of the modulus of continuity are obtained with respect to some particular possibility measures. For some subclasses of functions we prove that these Landau-Choquet type operators can have essentially better approximation properties than their classical correspondents.

Integral Representation of Coherent Lower Previsions by Super-Additive Integrals

Coherent lower previsions generalize the expected values and they are defined on the class of all real random variables on a finite non-empty set. Well known construction of coherent lower previsions by means of lower probabilities, or by means of super-modular capacities-based Choquet integrals, do not cover this important class of functionals on real random variables. In this paper, a new approach to the construction of coherent lower previsions acting on a finite space is proposed, exemplified and studied. It is based on special decomposition integrals recently introduced by Even and Lehrer, in our case the considered decomposition systems being single collections and thus called collection integrals. In special case when these integrals, defined for non-negative random variables only, are shift-invariant, we extend them to the class of all real random variables, thus obtaining so called super-additive integrals. Our proposed construction can be seen then as a normalized super-additive integral. We discuss and exemplify several particular cases, for example, when collections determine a coherent lower prevision for any monotone set function. For some particular collections, only particular set functions can be considered for our construction. Conjugated coherent upper previsions are also considered.

On outer measures and semi-separation of lattices

This present paper is concerned with set functions related to{0,1}two valued measures. These set functions are either outer measures or have many of the same characteristics. We investigate their properties and look at relations among them. We note in particular their association with the semi-separation of lattices.To be more specific, we define three set functionsμ″,μ′, andμ˜related toμ ϵ I(L)the{0,1}two valued set functions defined on the algebra generated by the lattice of setsL st μis a finitely additive monotone set function for whichμ(ϕ)=0. We note relations among them and properties they possess.ln particular necessary and sufficient conditions are given for the semi-separation of lattices in terms of equality of set functions over a lattice of subsets.Finally the notion ofI-lattice is defined, we look at some properties of these with certain other side conditions assume, and end with an application involving semi-separation andI-lattices.