I have doubts on the meaning of the symbol $L_r(P)$ for $r\in \mathbb{N}$. Consider a random variable $X: \Omega \rightarrow \mathbb{R}$ with probability distribution $P$ and a random function $f:\mathcal{X}\rightarrow \mathbb{R}$. So far I have encountered $L_r(P)$ in these occasions:
(1) $L_r(P)$-norm: $\|f\|_{P,r}:=\Big(\int_{\mathcal{X}}(f(x))^rdP(x)\Big)^{\frac{1}{r}}$
Question: should the $(\cdot)^{\frac{1}{r}}$ be there? From here it seems no, but I found other sources (e.g. van der Vaart "Asymptotic Statistics" p.270) in which it is there.
(2) $L_r(P)$ as a class of functions for which the $L_r(P)$-norm exists and is finite. Correct?
(3) $L_r(P)$-metric: $\|f-g\|_{P,r}:=\Big(\int_{\mathcal{X}}|f(x)-g(x)|^rdP(x)\Big)^{\frac{1}{r}}$. Correct?