Some relations between Hewitt-Stromberg premeasure and Hewitt-Stromberg measure

Abstract

Let $K$ be a compact set of $\R^n$ and $t\ge 0$. In this paper, we discuss the relation between the $t$-dimensional Hewitt-Stromberg premeasure and measure denoted by $\overline \HH^t$ and $\HH^t$ respectively. We prove : if $\overline{\HH}^t(K) $ $=+\infty$ then $\overline{\HH}^t(K)= {\HH}^t(K)$ and if $\overline{\HH}^t(K) =+\infty$, there exists a compact subset $F$ of $K$ such that $\overline{\HH}^t(F)= {\HH}^t(F)$ and ${\HH}^t(F)$ is close as we like to ${\HH}^t(K)$.