On the spaceability of the set of functions in the Lebesgue space $L_p$ which are in no other $L_q$

Abstract

In this note we prove that, for $p>0$, $L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]$ is $(\alpha,\mathfrak{c})$-spaceable if, and only if, $\alpha<\aleph_{0}$. Such a problem first appears in [V. F\'avaro, D. Pellegrino, D. Tomaz, Bull. Braz. Math. Soc. \textbf{51} (2020) 27-46], where the authors get the $(1,\mathfrak{c})$-spaceability of $L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]$ for $p>0$. The definitive answer to this problem continued to be sought by other authors, and some partial answers were obtained. The veracity of this result was expected, as a similar result is known for sequence spaces.