This is an announcement for the paper "Subspaces of maximal dimension contained in $L_{p}(\Omega) - \textstyle\bigcup\limits_{q<p}L_{q}(\Omega)$}" by G. Botelho, D. Cariello, V.V. Favaro, D. Pellegrino and J.B. Seoane-Sepulveda. Abstract: Let $(\Omega,\Sigma,\mu)$ be a measure space and $1< p < +\infty$. In this paper we determine when the set $L_{p}(\Omega) - \bigcup\limits_{1 \leq q < p}L_{q}(\Omega)$ is maximal spaceable, that is, when it contains (except for the null vector) a closed subspace $F$ of $L_{p}(\Omega)$ such that $\dim(F) = \dim\left(L_{p}(\Omega)\right)$. The aim of the results 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 $L_{p}(\Omega) - \bigcup\limits_{1 \leq q < p}L_{q}(\Omega)$. We shall also give examples, propose open questions and provide new directions in the study of maximal subspaces of classical measure spaces. Archive classification: math.FA Submitted from: dmpellegrino@gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2170 or http://arXiv.org/abs/1204.2170