  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.
