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 $p>0$, $L_p[0,1]\setminus {\bigcup }_{q\in (p,\infty )}{L}_q[0,1]$ is $(\alpha ,\mathfrak{c})$-spaceable if, and only if, $\alpha <{\aleph }_0$. Such a problem first appears in [V. Fávaro, D. Pellegrino, D. Tomaz, Bull. Braz. Math. Soc. \textbf{51} (2020) 27-46], where the authors get the $(1,\mathfrak{c})$-spaceability of $L_p[0,1]\setminus {\bigcup }_{q\in (p,\infty )}{L}_q[0,1]$ 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.