arXiv subjects

Piotr Mikusinski

Publications and source records attributed to Piotr Mikusinski.

5 recordsLinked to original sources

arXiv2019

A characterization of the Radon-Nikodym property for vector valued measures

If $μ_1,μ_2,\dots$ are positive measures on a measurable space $(X,Σ)$ and $v_1,v_2, \dots$ are elements of a Banach space ${\mathbb E}$ such that $\sum_{n=1}^\infty \|v_n\| μ_n(X) < \infty$, then $ω(S)= \sum_{n=1}^\infty v_n μ_n(S)$ defines a vector measure of bounded variation on $(X,Σ)$. We show ${\mathbb E}$ has the Radon-Nikodym property if and only if every ${\mathbb E}$-valued measure of bounded variation on $(X,Σ)$ is of this form. As an application of this result we show that under natural conditions an operator defined on positive measures, has a unique extension to an operator defined on ${\mathbb E}$-valued measures for any Banach space ${\mathbb E}$ that has the Radon-Nikodym property.

math.FA↗

arXiv2018

Pseudoquotient extensions of measure spaces

A space of pseudoquotients $\mathcal P (X,S)$ is defined as equivalence classes of pairs $(x,f)$, where $x$ is an element of a non-empty set $X$, $f$ is an element of $S$, a commutative semigroup of injective maps from $X$ to $X$, and $(x,f) \sim (y,g)$ if $gx=fy$. In this note we assume that $(X,Σ,μ)$ is a measure space and that $S$ is a commutative semigroup of measurable injections acting on $X$ and investigate under what conditions there is an extension of $μ$ to $\mathcal P (X,S)$.

math.CA↗

arXiv2016

The Daniell Integral

The basic properties of the Daniell integral are presented. We do not use the standard approach of introducing auxiliary spaces of the "over-functions" and "under-functions." Instead, we use a simple and direct approach based on approximating integrable functions by absolutely convergent series of simple functions.

math.CA↗

arXiv2015

Integrals with values in Banach spaces and locally convex spaces

The purpose of this article is to present the construction and basic properties of the general Bochner integral. The approach presented here is based on the ideas from the book The Bochner Integral by J. Mikusinski where the integral is presented for functions defined on $\mathbb{R}^N$. In this article we present a more general and simplified construction of the Bochner integral on abstract measure spaces. An extension of the construction to functions with values in a locally convex space is also considered.

math.FA↗

arXiv2011

A Sheaf of Boehmians

We show that Boehmians defined over open sets of $\mathbb{R}^N$ constitute a sheaf. In particular, it is shown that such Boehmians satisfy the gluing property of sheaves over a topological space.

math.FA↗