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

For each $b\in (0,\mathrm{\infty })$ we intend to generate a decreasing sequence of subsets $({\mathcal{Y}}_b^{(n)})\subset Y_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}$ 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{c}\mathrm{o}\mathrm{n}\mathrm{c}$}$ are pairwise equinumerous. Among others we also show that if $f$ is any measurable function on a measure space $(\mathrm{\Omega },\mathcal{F},\lambda )$ and $p\in[ 1,\infty) $isanarbitrarynumberthenthequantities$\left\Vert f\right\Vert_{L^{p}}$and$\sup_{\Phi\in\widetilde{$\mathcal{Y}${$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}$}}}(\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.