A Note on Fractal Measures and Cartesian Product Sets

Abstract

In this paper, we give a new product formula :where E ⊆ R d , F ⊆ R l , t, s ≥ 0 and H t and P s denote, respectively, the lower and upper Hewitt-Stromberg measures. Using these inequalities, we give lower and upper bounds for the lower and upper Hewitt-Stromberg dimensions b(E × F) and B(E × F) in terms of the Hewitt-Strombeg dimensions of E and F.

A note on the generalized Hausdorff and packing measures of product sets in metric space

“…H q,h µ (E) P q,g ν (F ) ≤ cP q,hg µ×ν (E × F ) (1.5) P q,hg µ×ν (E × F ) ≤ cP q,h µ (E) P q,g ν (F ), (1.6) provided that µ and ν satisfy the doubling condition. Similar results were be proved for the Hewitt-Stromberg measures [1,10] ( see [11,12,9] for more details on these measures).…”

On a class of Hausdorff measure of cartesian product sets

“…for any E ⊆ X and F ⊆ X ′ . This result has been proved in [31] for any subsets E and E ′ of Euclidean spaces R d (d ≥ 1) provided that µ and ν are blanketed measures and in [2] by investigating the density result introduced in [29]. The disadvantage of density approach includes the inability to handle sets of measure ∞ .…”