Let $G$ be a compact Hausdorff group and let $\int$ be its Haar integral.
I am interested in the difference between $L_p$ spaces in both the compact and discrete settings (discrete groups)
For the compact case I think there is no difference, right?
For a Borel measure on a compact Hausdorff topological space, it seems like $L_p$ and $L_r$ space are the same, since $$\int |f|^p ≤ \int |c_{\max f}|^p$$ for $f$ a non-negative continuous function, where $c_{\max f}$ is the maximum value attained by $f$. Is this true?
For instance, isn't it the case that $L_1(S^1) ≅ L_2(S^1)$?
Oh wait, I think I understand where I go wrong- we mod out to get Lₚ space, so $\text{L}^{∞}$ actually quotients onto the rest in this case.
Meanwhile is something dual the case for the discrete groups and counting measures?