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

Abstract

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

Some minor improvements in the abstract and introduction that could otherwise be misleading. Besides the preciseness of statements, the definition of an $M$-sensitive random variable should include a regularity condition such as $\mathbb{P}$-essential boundedness. Some slight improvements in the proof are made accordingly. For v3: Two slight improvements are made in the proof, one for a remark