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

Abstract

We show that, for every $1 \leq p < +\infty$ and for every Borel probability measure $\mathbb{P}$ over $\mathbb{R}$, every element of $L^{p}(\mathbb{R}, \mathscr{B}_{\mathbb{R}}, \mathbb{P})$ is the $L^{p}$-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 $\mathbb{R}$.