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In this short note we prove the result stated in the title; that is, forevery $p>0$ there exists an infinite dimensional closed linear subspace of$L_{p}[0,1]$ every nonzero element of which does not belong to$\bigcup\limits_{q>p} L_{q}[0,1]$. This answers in the positive a questionraised in 2010 by R. M. Aron on the spaceability of the above sets (for both,the Banach and quasi-Banach cases). We also complete some recent results from\cite{BDFP} for subsets of sequence spaces.

$L_{p}[0,1] \setminus \bigcup\limits_{q>p} L_{q}[0,1]$ is spaceable for every $p>0$