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.
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