Bijections and metric spaces induced by some collective properties of concave Young-functions

Abstract

For each ${\small b\in(0, \infty)}$ we intend to generate a decreasing sequence of subsets $(\mathcal{Y}_{b}^{(n)}) \subset Y_{\mathrm{conc}}$ depending on $b$ such that whenever $n\in\mathbb{N}$, then $\mathcal{A}\cap\mathcal{Y}_{b}^{(n)}% $ is dense in $\mathcal{Y}_{b}^{(n)}$ and the following four sets $\mathcal{Y}_{b}^{(n)}$, $\mathcal{Y}_{b}^{(n) }\backslash(\mathcal{A}\cap\mathcal{Y}_{b}^{(n)}) $, $\mathcal{A}\cap\mathcal{Y}_{b}^{(n)}$ and $\mathcal{Y}_{\mathrm{conc}}$ are pairwise equinumerous. Among others we also show that if $f$ is any measurable function on a measure space $(\Omega,\mathcal{F},\lambda) $ and $p\in[ 1,\infty) $ is an arbitrary number then the quantities $\left\Vert f\right\Vert_{L^{p}}$ and $\sup_{\Phi\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}}(\Phi(1)) ^{-1}\left\Vert \Phi\circ| f| \right\Vert_{L^{p}}$ are equivalent, in the sense that they are both either finite or infinite at the same time.