Let's say $\mathcal{A}$ is some subset of $\mathcal{M}_{N}(\mathbb{R})$ (square matrices of order $N$ with real coefficients).
What does $L^{p}(\mathbb{R}^{N},\mathcal{A})$ mean (where $p\in\mathbb{N}\cup\{\infty\}$)? I guess it means the set of $L^{p}$-functions from $\mathbb{R}^{N}$ to $\mathcal{A}$, but with which norm is it equipped and what does that even mean? (We consider the Lebesgue measure and the $\sigma$-algebra of Lebesgue measurable subsets of $\mathbb{R}^{N}$)
EDIT: would it be equipped with the following norm?
$$L^{p}(\mathbb{R}^{N},\mathcal{A})\to\mathbb{R}^{+}:f\to\left[\int_{\mathbb{R}^{N}}\Vert f\Vert^{p}_{\mathcal{A}}\text{d}\mu\right]^{1/p}$$
Any help is appreciated.