I just started learning about $L^p$ spaces today and I have this question: Let $(X,\scr{M},\mu)$ be a measure space. Let $f:X\rightarrow \mathbb{C}$ be measurable. Consider $\eta:]0,\infty[\rightarrow[0,\infty]$ that sends $p$ to $||f||_p$. Is $\eta$ continuous ?
My approach was to show that the function $p\rightarrow \int_X |f|^P d\mu$ is continuous by showing that for every sequence $\{p_n\}$ that converges to $p$ we have $\int_X |f|^{p_n}d\mu$ converges to $\int_X |f|^p d\mu$ using Lebesgue's dominated convergence theorem, but I couldn't as I wasn't able to find a dominating function. I also think that the claim of my approach is stronger than the claim "$\eta$ is continuous".
Thank you
