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Kazimierz Musial

Publications and source records attributed to Kazimierz Musial.

9 recordsLinked to original sources
arXiv2026

Splitting of Liftings in Product Spaces II

Let $(X, \mfA,P)$ and $(Y, \mfB,Q)$ be two probability spaces, $R$ be their skew product on the product $\sigma$-algebra $\mfA\otimes\mfB$ and $\{(\mfA_y,S_y)\colon y\in{Y}\}$ be a $Q$-disintegration of $R$. Then let $\mfA\dd\mfB$ be the $\sigma$-algebra generated $\mfA\otimes\mfB$ and by the family $\mcM:=\{E\subset{X\times{Y}}\colon \exists\;N\in\mfB_0\;\forall\;y\notin{N}\;\wh{S_y}(E^y)=0\}$ and $\wh{R_{\dd}}$ be the extension of $R$ such that $\mcM$ becomes the family of $\wh{R_*}$-zero sets ($\wh{S_y}$ is the completion of $S_y$ and $\mfB_0=\{B\in\mfB: Q(B)=0\}$). We prove that there exist a lifting $\pi$ on $\mcL^{\infty}(\wh{R_{\dd}})$ and liftings $\sigma_y$ on $\mcL^{\infty}(\wh{S_y})$ , $y\in Y$, such that \[ [\pi(f)]^y= \sigma_y\Bigl([\pi(f)]^y\Bigr) \qquad\mbox{for every} \quad y\in Y\quad\mbox{and every}\quad f\in\mcL^{\infty}(\wh{R_{\dd}}). \] In case of a separable $P$ and in case when $R\ll{P}\times{Q}$ a characterization of stochastic processes possessing an equivalent measurable version is presented. The theorem is a generalization and correction of \cite[Theorem 3.8]{mu25}.

math.PR↗
arXiv2025

Convergence for varying measures

Some limit theorems of the type $\int_Ωf_n dm_n -- --> \int_Ωf dm$ are presented for scalar, (vector), (multi)-valued sequences of m_n-integrable functions f_n. The convergences obtained, in the vector and multivalued settings, are in the weak or in the strong sense.

math.FA↗
arXiv2023

Conditional Expectations in Banach spaces with RNP

Let $X$ be a Banach space with RNP, $(\vO,\vS,μ)$ be a complete probability space and $\vG:\vO\to{cb(X)}$ (nonempty, closed convex and bounded subsets of $X$) be a multifunction. Assume that $\vX\subset\vS$ is a $σ$-algebra and the multimeasure $M$ defined by the Pettis integral of $\vG$ be such that the restriction of $M$ to $\vX$ is of $σ$-finite variation. Using a lifting, I prove the existence of an Effros measurable conditional expectation of $\vG$ and present its representation in terms of quasi-selections of $\vG$. I apply then the description to martingales of Pettis integrable multifunctions obtaining a scalarly equivalent martingale of measurable multifunctions with many martingale selections. In general the situation cannot be reduced to the separable space.

math.FA↗
arXiv2023

Splitting of Liftings in Product Spaces

Let $(X, {\mathfrak A},P)$ and $(Y, {\mathfrak B},Q)$ be two probability spaces and $R$ be their skew product on the product $σ$-algebra ${\mathfrak A}\otimes\mfB$. Moreover, let $\{({\mathfrak A}_y,S_y)\colon y\in{Y}\}$ be a $Q$-disintegration of $R$ (if ${\mathfrak A}_y={\mathfrak A}$ for every $y\in{Y}$, then we have a regular conditional probability on ${\mathfrak A}$ with respect to $Q$) and let $\mfC$ be a sub-$σ$-algebra of ${\mathfrak A}\cap\bigcap_{y\in{Y}}{\mathfrak A}_y$. For $f\in\mcL^{\infty}(R)$ I investigate the relationship between the $Y$-sections $[{\mathbb E}_{\mfC\otimes\mfB}(f)]^y$ of ${\mathbb E}_{\mfC\otimes\mfB}(f)$ (the conditional expectation of $f$ with respect to $\mfC\otimes\mfB$) and the conditional expectations of $f^y$ with respect $\mfC$ and $S_y$. Moreover I prove the existence of a lifting $π$ on $\mcL^{\infty}(\wh{R})$ ($\wh{R}$ is the completion of $R$) and liftings $σ_y$ on $\mcL^{\infty}(\wh{S_y})$, $y\in Y$, such that \begin{equation*} [π(f)]^y= σ_y\Bigl([π(f)]^y\Bigr) \qquad\mbox{for all} \quad y\in Y\quad\mbox{and}\quad f\in\mcL^{\infty}(\wh{R}). \end{equation*} As an application a characterization of stochastic processes possessing an equivalent measurable version is presented.

math.FA↗
arXiv2023

Representations of multimeasures via multivalued Bartle-Dunford-Schwartz integral

An integral for a scalar function with respect to a multimeasure $N$ taking its values in a locally convex space is introduced. The definition is independent of the selections of $N$ and is related to a functional version of the Bartle-Dunford-Schwartz integral with respect to a vector measure presented by Lewis. Its properties are studied together with its application to Radon-Nikodym theorems in order to represent as an integrable derivative the ratio of two general multimeasures or two $d_H$-multimeasures; equivalent conditions are provided in both cases.

math.FA↗
arXiv2020

Relations among Gauge and Pettis integrals for $cwk(X)$-valued multifunctions

The aim of this paper is to study relationships among "gauge integrals" (Henstock, Mc Shane, Birkhoff) and Pettis integral of multifunctions whose values are weakly compact and convex subsets of a general Banach space, not necessarily separable. For this purpose we prove the existence of variationally Henstock integrable selections for variationally Henstock integrable multifunctions. Using this and other known results concerning the existence of selections integrable in the same sense as the corresponding multifunctions, we obtain three decomposition theorems. As applications of such decompositions, we deduce characterizations of Henstock and ${\mathcal H}$ integrable multifunctions, together with an extension of a well-known theorem of Fremlin.

math.FA↗
arXiv2019

Integration of multifunctions with closed convex values in arbitrary Banach spaces

Integral properties of multifunctions with closed convex values are studied. In this more general framework not all the tools and the technique used for weakly compact convex valued multifunctions work. We pay particular attention to the "positive multifunctions". Among them an investigation of multifunctions determined by vector-valued functions is presented. Finally, decomposition results are obtained for scalarly and gauge-defined integrals of multifunctions and a full description of McShane integrability in terms of Henstock and Pettis integrability is given.

math.FA↗