Sensitive Random Variables are Dense in Every $L^p(\mathbb{R},{\mathcal{B}}_{\mathbb{R}},\mathbb{P})$

We show that, for every $1\le p<+\mathrm{\infty }$ and for every Borel probability measure $\mathbb{P}$ over $\mathbb{R}$, every element of $L^{p}($\mathbb{R}$, \mathscr{B}_{$\mathbb{R}$}, $\mathbb{P}$)$isthe$L^{p}$-limit of some sequence of bounded random variables that are Lebesgue-almost everywhere differentiable with derivatives having norm greater than any pre-specified real number at every point of differentiability. In general, this result provides, in some direction, a finer description of an $L^p$-approximation for $L^p$ functions on $\mathbb{R}$.