Most often (at least in probability), one defines the $L^p$ space as
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $p\geq 1$ be a real number. Then $$ L^p(\Omega, \mathcal{F}, \mathbb{P}) := \left\{f \left| f:\Omega\to\mathbb{R} \text{ is } \mathcal{F}\text{-measurable and } \int |f(\omega)|^p \mathbb{P}(d\omega) < \infty\right.\right\}. $$
Is it possible to generalise this to $(\Omega, \mathcal{F}, \mathbb{M})$ being a measure space (not necessarily sigma-finite) and $X:\Omega\to\mathsf{E}$ be a random variable taking values in $\mathsf{E}$ which is now a Banach space with norm $|\cdot|_\mathsf{E}$ and where $(\mathsf{E}, \mathcal{E})$ is a measurable space? For instance:
Let $(\Omega, \mathcal{F}, \mathbb{M})$ be a measure space, $(\mathsf{E}, \mathcal{E})$ be a measurable space with $(\mathsf{E}, |\cdot|_\mathsf{E})$ being a Banach space. Let $X:\Omega\to\mathsf{E}$ be a random variable and $p \geq 1$ be real. Then we define $$ L^p(\Omega, \mathcal{F}, \mathbb{M}) := \left\{f \left| f:\Omega\to\mathsf{E} \text{ is } \mathcal{F}\text{-measurable and } \int |f(\omega)|^p_\mathsf{E} \mathbb{M}(d\omega) < \infty\right.\right\}. $$