Sensitive Random Variables are Dense in Every L^p(ℝ, \mathscrB_ℝ, ℙ)

2020/06/12 by Yu-Lin Chou, Chou, Yu-Lin
Economics, Econometrics and Finance · Mathematics · #26A24 #46E30 #60A10 #Advanced Harmonic Analysis Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2006.07372

openalex publication_date 2020/06/12 · openalex created_date 2020/06/19 · openalex updated_date 2026/07/28

Abstract

We show that, for every 1 ≤ p < +∞ and for every Borel probability measure ℙ over ℝ, every element of L^p(ℝ, \mathscrB_ℝ, 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 ℝ.

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