I've recently had a look at https://helios2.mi.parisdescartes.fr/~jdedecke/p1.pdf . In chapter 3, Definition 3.1, they defined: For any $p \geq 1$, let $\mathbb{L}^p$ be the class of real-valued function of $(\mathcal{X},P)$ ($\mathcal{X}$ is a Polish space and $P$ is a (probability) measure) such that $\lVert f \rVert_p^p := P(\lvert f \rvert^p)$ is finite.
My question: Is the definition of the $\lVert \cdot \rVert_p$-norm equivalent to the normal one, i.e. does it hold that $(P(\lvert f \rvert^p))^{1/p}= (\mathbb{E}(\lvert f \rvert^p))^{1/p}$? Is the class $\mathbb{L}^p$ the same as the $\mathcal{L}^p$ spaces? (I think the difference ist, that $\mathcal{L}^p$ are measurable functions, but not necessarily real-valued and otherwise $\mathbb{L}^p$ are real-valued but not necessarily measurable functions). Or if not, are there cases where they coincide?
Thanks for any help. :)