Metrization of a Space of Weakly Additive Functionals

We investigate a space of weakly additive, order-preserving, normed and positively homogeneous functionals on a metric compact. We construct an analog of modified Kantorovich–Rubinshtein metric on a space O(X) of weakly additive normed functionals on a metric compact X.     

Open mapping theorem for spaces of weakly additive homogeneous functionals

We establish that if X and Y are metric compacta and f: X → Y is a continuous surjective mapping, then the openness of the mapping OH(f): OH(X) → OH(Y) of spaces of weakly additive homogeneous functionals is equivalent to the openness of f.     

Representation of martingale additive functionals on Banach spaces

Assume thatB is a separable real Banach space andX(t) is a diffusion process onB. In this paper, we will establish the representation theorem of martingale additive functionals ofX(t).     

Some categorical properties of the functors O τ and O R of weakly additive functionals

In the paper, the spaces of weakly additive τ-smooth and Radon functionals are investigated. It is proved that the functors of weakly additive τ-smooth and Radon functionals weakly preserve the density of Tychonoff spaces, and the functor of weakly additive τ-smooth functionals forms a monad in the category of Tychonoff spaces and their continuous mappings. Examples and remarks are given showing that these functors fail to satisfy certain Shchepin normality conditions. Problems having positive solutions for normal functors are presented.     

On certain properties of the spaces of order-preserving functionals

In the present paper it is proved that the functor Oτ of τ-smooth order preserving functionals and the functor OR of Radon order preserving functionals, do not change the weight of infinite Tychonoff spaces. It is shown that the density and the weak density of infinite Tychonoff spaces do not increase under these functors. Moreover, if X is a metric space with the second axiom of countability then the spaces Oτ(X) and OR(X) are also metrizable.     

The positive aspects of smoothness in Banach lattices

Let X be a Banach lattice, and let ${x\in X{\setminus}\{0\}}$ . We study the structure of the set Grad(x), of all supporting functionals of x. If X is a Dedekind σ-complete Banach lattice, there is an isometry from Grad(x) onto Grad(|x|); hence the elements x and |x| are smooth simultaneously. And if, additionally, X* is strictly monotone then Grad(|x|) consists of positive functionals. As a by-product of our results we obtain that an arbitrary Banach lattice X is strictly monotone whenever its dual X* is smooth.     

Yet on linear structures of norm-attaining functionals on Asplund spaces

In this paper, we show that if an Asplund space X is either a Banach lattice or a quotient space of C(K), then it can be equivalently renormed so that the set of norm-attaining functionals contains an infinite dimensional closed subspace of X^* if and only if X^* contains an infinite dimensional reflexive subspace, which gives a partial answer to a question of Bandyopadhyay and Godefroy.     

Sure wins, separating probabilities and the representation of linear functionals

We discuss conditions under which a convex cone K⊂RΩ admits a finitely additive probability m such that supk∈Km(k)?0. Based on these, we characterise those linear functionals that are representable as finitely additive expectations. A version of Riesz decomposition based on this property is obtained as well as a characterisation of positive functionals on the space of integrable functions.     

Optimal approximation and quantisation

A number of interesting functionals F(X) of a finite point set X in the Euclidean space can be represented as integrals of a function η that depends on the integration variable y and X restricted onto a certain set C(y,X) that is determined by y and X and satisfies separation and uniform boundedness conditions. For instance, C(y,X) can be the Voronoi cell generated by X that contains y. We single out the general properties of C and η that ensure that the normalised infimum of F(X) over all sets X with cardinality n converges to a limit that can be identified using a particular form of η. The considered functionals include those that appear in quantisation problems for probability measures, and in finding the optimal approximation of a function by splines, tangent planes and triangulated surfaces. For instance, it is shown that the minimum Lβ-approximation error (normalised by nβ) of a sufficiently smooth bivariate convex function f using the convex triangulation converges to the L1/(β+1)-norm of the determinant of the second derivative of f times a certain absolute constant.     

Graphical convergence of continuous functions

Let X and Y be metrizable spaces. We show that convergence of a net of continuous functions 〈f _λ 〉 to a continuous function f in the graph topology for C(X,Y) is equivalent to the uniform convergence of the net of associated distance functionals for the graphs with respect to each compatible metric on X×Y. Remarkably, no weaker convergence results if uniform convergence is replaced by pointwise convergence in the last statement.     

On the limit laws of the second order for additive functionals of Harris recurrent Markov chains

Let {X _n } _{n ≥0} be a Harris recurrent Markov chain with state space E and invariant measure π. The law of the iterated logarithm and the law of weak convergence are given for the additive functionals of the form

A Survey on the Pseudo-process Driven by the High-order Heat-type Equation \boldsymbol{\partial/\partial t=\pm\partial^N\!/\partial x^N} Concerning the Hitting and Sojourn Times

Fix an integer N?>?2 and let X?=?(X(t)) _t????0 be the pseudo-process driven by the high-order heat-type equation $\partial/\partial t=\pm\partial^N\!/\partial x^N$ . The denomination ??pseudo-process?? means that X is related to a signed measure (which is not a probability measure) with total mass equal to one. In this survey, we present several explicit results and discuss some problems concerning the pseudo-distributions of various functionals of the pseudo-process X: the first or last overshooting times of a single barrier {a} or a double barrier {a, b} by X; the sojourn times of X in the intervals [a,?+????) and [a, b] up to a fixed time; the maximum or minimum of X up to a fixed time.     

Extension theorems of Hahn-Banach type for nonlinear disjointly additive functionals and operators in Lebesgue spaces

Let (Ω, τ, M) be a nonatomic separable finite measure space. Every continuous functional N on L^p(m), 1 ? p < ∞, which is disjointly additive in the sense N(u + v) = N(u) + N(v) whenever uv = 0, is known to be representable by an integral with a nonlinear Caratheodory kernel. Such functionals share several regularity properties with continuous linear functionals. Here we study the question of whether every continuous, disjointly additive functional defined on a closed subspace of L^p(m) possesses an extension to L^p(m) with these same properties. This question has applications to the study of nonlinear functionals on Sobolev spaces. It is shown that for a class of subspaces, including those of finite codimension, such an extension always exists, but there are also closed subspaces not possessing this extension property. Analogous results are obtained for disjointly additive mappings from closed subspaces of L^p(m) into L¹(m) and for functionals defined on subspaces of L^∞(m). The techniques depend heavily on the utilization of Lyapunov vector measures.     

An abstract Radon-Nikodym theorem

In this paper we obtain a Radon-Nikodym theorem for positive linear functionals on a B¹-algebra M. Some corollaries analogous to those obtained in the classical case are also obtained here. It is known that if X is a Banach space, then the space $\text{L}{}_1\text{(Ω, X)}$ of Bochner integrable functions on a probability space Ω with values in X is the completion (in a suitable topology) of the tensor product $\text{L}{}_1\text{(Ω) ? X}$. Using our theorem, it is possible to extend this result for certain linear mappings from M ? X to X.     

Additive local spectrum compressors

Let L(X) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. We characterize additive continuous maps from L(X) onto itself which compress the local spectrum and the convexified local spectrum at a nonzero fixed vector. Additive continuous maps from L(X) onto itself that preserve the local spectral radius at a nonzero fixed vector are also characterized.     

Best approximation in vector spaces

In this paper, we obtain some results on best f-approximation in quotient spaces of vector spaces and determine under what conditions f-proximinality can be transmitted to and from quotient spaces. Also, in conclusion, we consider the relationship between f-approximation subsets and linear functionals on X.     

On certain spaces of vector measures of bounded variation

If (Σ,X) is a measurable space and X a Banach space we investigate the X-inheritance of copies of ?_∞ in certain subspaces Δ(Σ,X) of bvca(Σ,X), the Banach space of all X-valued countable additive measures of bounded variation equipped with the variation norm. Among the consequences of our main theorem we get a theorem of J. Mendoza on the X-inheritance of copies of ?_∞ in the Bochner space L₁(μ,X) and other of the author on the X-inheritance of copies of ?_∞ in bvca(Σ,X).     

The Average Range Characterization of the Radon–Nikodym Property

A characterization of Banach spaces possessing the Radon—Nikodym property is given in terms of the average range of additive interval functions. We prove that a Banach space X has the RNP if and only if each X-valued additive interval function possessing absolutely continuous McShane (or Henstock) variational measure has nonempty average range almost everywhere on [0, 1].