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 Lebesguealmost everywhere differentiable with derivatives having norm greater than any prespecified 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 $\mathbb{R}$.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 DOI:
 10.48550/arXiv.2006.07372
 arXiv:
 arXiv:2006.07372
 Bibcode:
 2020arXiv200607372C
 Keywords:

 Mathematics  Probability;
 60A10;
 46E30;
 26A24
 EPrint:
 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