Integrals for functions with values in a partially ordered vector space

We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure space leads to the classical space of integrable functions

Bochner integrals in ordered vector spaces

We present a natural way to cover an Archimedean directed ordered vector space $E$ by Banach spaces and extend the notion of Bochner integrability to functions with values in $E$. The resulting set of integrable functions is an Archimedean directed ordered vector space and the integral is an order preserving map

The Measure-Theoretical Approach to P-Adic Probability Theory

this paper we develop a p-adic probability formalism based on measure theory of [19]. By probabilistic reasons we use the special case of this measure theory: measures defined on algebras (such measures have some special properties). However, probabilistic applications stimulate also the development of the general theory of non-Archimedean measures defined on rings. We prove the formula of the change of variables for these measures and use this formula for developing the formalism of conditional expectations fo

Convergence in the Hausdorff metric of estimators of irregular densities, using Fourier-Cesàro approximation

The problem of estimating a density which is allowed to have discontinuities of the first kind is considered. The usual Fourier-type estimator is based on the Dirichlet or sine kernel and is not suitable to eliminate the Gibbs phenomenon. Fourier-Cesàro approximation yields the Fejér kernel which is the square of the sine function. Density estimators based on the Fejér kernel do control the Gibbs phenomenon. Integral metrics are not sufficiently sensitive to properly assess the performance of estimators of irregular signals. Therefore, we use the Hausdorff distance between the extended, closed, graphs of estimator and estimand, and derive an a.s. speed of convergence of this distance.Density estimation Irregular densities Gibbs phenomenon Fejer kernel Hausdorff metric