Suppose $f$ is Lebesgue measurable on $(0,1)$, and not essentially bounded. Can $||f||_p$ tend to $\infty$ arbitrarily slowly?
I am really lost on this problem. I assume the answer is that it can tend to $\infty$ arbitrarlily slowly but showing that is really difficult. I have tried letting $f=\sum\limits_{k=1}^{\infty}a_k 1_{E_k}$ where $E_i$ is a disjoint collection of subset of $(0,1)$, and working from there but I am at a loss here. Any guidence?
Arbitrarilly slowly- We say that $||f||_p$ tend to infinity arbitrarilly slowly if for every positive function $\Phi(p)$ on $(0,\infty)$ with $\lim_{p\to \infty}\Phi(p)=\infty$ there exists a function $f$ on $(0,1)$ such that $||f||_p\leq \Phi(p)$ for all large enough $p$.