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Let $(\Omega,\Sigma,\mu)$ be a measure space and $1< p < +\infty$. In thispaper we show that, under quite general conditions, the set $L_{p}(\Omega) -\bigcup\limits_{1 \leq q < p}L_{q}(\Omega)$ is maximal spaceable, that is, itcontains (except for the null vector) a closed subspace $F$ of $L_{p}(\Omega)$such that $\dim(F) = \dim(L_{p}(\Omega))$. We also show that if thoseconditions are not fulfilled, then even the larger set $L_p(\Omega) -L_q(\Omega)$, $1 \leq q < p$, may fail to be maximal spaceable. The aim of theresults presented here is, among others, to generalize all the previous work(since the 1960's) related to the linear structure of the sets $L_{p}(\Omega) -L_{q}(\Omega)$ with $q < p$ and $L_{p}(\Omega) - \bigcup\limits_{1 \leq q <p}L_{q}(\Omega)$.

Subspaces of maximal dimension contained in $L_p(\Omega) - \bigcup\limits_{q