Optimal Banach function space generated with the cone of nonnegative increasing functions

The article deals with the effective constructions for the optimal Banach ideal and symmetric spaces (of functions $f:~[0,T]\to\mathbb{R}$) containing a cone of nonnegative and increasingly monotone functions with respect to the natural functional generated $L_p$-norm ($1\le p<\infty$). The first of these spaces turns out to be the space of measurable functions $f$ such that $\|f\|_{L_\infty(\cdot,T)}\in L_p(0,T)$; this space can be endowed with the norm $\|\,\|f\|_{L_\infty(\cdot,T)}\|f\|_{L_p(0,T)}$. The second coincides with the usual space $L_p$.