# A note on the Regularities of Hewitt-Stromberg h-measures

A note on the Regularities of Hewitt-Stromberg h-measures In this paper, we investigate the Hewitt-Stromberg h-measures Hh\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathsf {H}}^h$$\end{document} and Ph\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathsf {P}}^h$$\end{document} which lie between the Hausdorff h-measure Hh\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathcal {H}}}^h$$\end{document} and packing h-measure Ph\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathcal {P}}}^h$$\end{document} where h is a dimension function. We formulate a new version of density theorem given by Raymond and Tricot (1988) and then set up a necessary and sufficient condition for which these measures are equivalent. Also we study the notion of weak and strong regularity of sets. As an application we study the Hewitt-Stromberg measures of cartesian product sets by means of the measure of their components. We prove also that if, 0<Ht(E)=Pt(E)<+∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0<{\mathsf {H}}^t(E)={\mathsf {P}}^t(E) <+\infty$$\end{document}, then t∈N\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$t\in {\mathbb {N}}$$\end{document}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ANNALI DELL UNIVERSITA DI FERRARA Springer Journals

# A note on the Regularities of Hewitt-Stromberg h-measures

, Volume 69 (1) – May 1, 2023
17 pages

/lp/springer-journals/a-note-on-the-regularities-of-hewitt-stromberg-h-measures-A13BezlAbT

# References (42)

Publisher
Springer Journals
Copyright © The Author(s) under exclusive license to Università degli Studi di Ferrara 2022. corrected publication 2022
ISSN
0430-3202
eISSN
1827-1510
DOI
10.1007/s11565-022-00405-w
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we investigate the Hewitt-Stromberg h-measures Hh\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathsf {H}}^h$$\end{document} and Ph\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathsf {P}}^h$$\end{document} which lie between the Hausdorff h-measure Hh\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathcal {H}}}^h$$\end{document} and packing h-measure Ph\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathcal {P}}}^h$$\end{document} where h is a dimension function. We formulate a new version of density theorem given by Raymond and Tricot (1988) and then set up a necessary and sufficient condition for which these measures are equivalent. Also we study the notion of weak and strong regularity of sets. As an application we study the Hewitt-Stromberg measures of cartesian product sets by means of the measure of their components. We prove also that if, 0<Ht(E)=Pt(E)<+∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0<{\mathsf {H}}^t(E)={\mathsf {P}}^t(E) <+\infty$$\end{document}, then t∈N\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$t\in {\mathbb {N}}$$\end{document}.

### Journal

ANNALI DELL UNIVERSITA DI FERRARASpringer Journals

Published: May 1, 2023

Keywords: Hausdorff measure; Packing measure; Hewitt-Stromberg measure; Regularities; Densities; Primary 28A78; 28A80