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

TL;DR

This paper demonstrates that in any $L^{p}$ space over the real line, every function can be approximated by bounded, differentiable random variables with derivatives exceeding any given magnitude, offering a refined approximation perspective.

Contribution

It establishes that all $L^{p}$ functions can be densely approximated by differentiable variables with arbitrarily large derivatives, advancing understanding of $L^{p}$-approximations.

Findings

Any $L^{p}$ function can be approximated by differentiable variables.

Differentiable approximations can have derivatives exceeding any specified bound.

Provides a finer characterization of $L^{p}$-approximations on $ eal$.

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 $\mathbb{R}$.