| Abstrakt: |
Let μ and v be two Borel probability measures on two separable metric spaces X and Y respectively. For h,g be two Hausdorff functions and q ∈ ℝ, we introduce and investigate the generalized pseudo-packing measure R q,h μ and the weighted generalized packing measure Qμq,h to give some product inequalities : Hμ × vq,hg (E × F) ≤ Hμ q,h (E) Rvq,g(F) ≤ Rμ × vq,hg(E × F) and Hμ × vq,hg (E × F) ≤ Qμ q,h (E) Pvq,g(F) for all E ⊆ X and F ⊆ Y, where H q,h μ and P q,h μ is the generalized Hausdorff and packing measures respectively. As an application, we prove that under appropriate geometric conditions, there exists a constant c such that Hμ×vq,hg(E × F) ≤ c Hμq,h (E)Pvq,g (F) Hμq,h(E) Pvq,g(F) ≤ c Pμq,hg ( × F) Pμ×vq,hg (E × F) ≤ c Pμq,h (E)Pvq,g (F). These appropriate inequalities are more refined than well know results since we do no assumptions on μ, v,h and g. [ABSTRACT FROM AUTHOR] |
