Let $\sigma\subset[0,1]$ be a measurable set with positive Lebesgue-measure. The set $$ \{ \psi\in L^\infty(\sigma) \mid \operatorname{ess inf} \vert\psi\vert >0 \} $$ is dense in $L^2(\sigma)$, but is it meagre/nonmeagre/comeagre?
My problem is that I do not really understand these concepts and I can't find any good source which applies these terms to the $L^p$ rooms. Can you give me some hints where to finde the answer or literature which adress this problem?
The motivation behind this question is that one can identify each possible frame $(T^n f)_{n\in\mathbb{Z}}$ (for a given operator $T$) in a Hilbert space with a specific $\psi$ from the upper set. Actually I want to analyse the set of $f$'s in that Hilbert space that form a frame. I think that the $L^p$ spaces are analysed quite well so that it would be easier to analyse the upper set.