Is there a generalized concept for an $L^{p}$-norm to be applicable to $\mathbb{R}^{n}$-valued functions for $n \geq 3$? I wonder if there is already an established "generalized" $L^{p}$-norm to be applicable to $\mathbb{R}^{n}$-valued functions for $n \geq 3$? The treatments I have seen seem to stop at $\mathbb{C}$-valued functions. If there is any intrinsic problem preventing developing in that direction, I would also be of interest about it. Thanks. By the way, please do not exclude the possibility that the OP simply missed a relatively obvious point here.
 A: Actually, there is a much more general theory. Let $(\Omega, \Sigma, \mu)$ be a measure space and $X$ be a Banach space. Then for a scalar function $f : \Omega \to \mathbb C$ and $x \in X$ one can define the function $f \otimes x: \Omega \to X,\ f \otimes x(t) = f(t) x$. Then one defines vector-valued simple functions to be elements like $\mathbf 1_A \otimes x$ where $A \in \Sigma$ and $x \in X$. 
Now a mapping $f: \Omega \to X$ is called $\mu$-measurable if there is a sequence of simple functions $(f_n)_{n \in \mathbb N}$ such that $f_n \to f$ almost everywhere in $X$. Another good criteria for when a function $f: \Omega \to X$ is $\mu$-measurable is the Pettis measurability theorem which is often applied to show that a certain function is $\mu$-measurable. 
For such functions one can define the Bochner integral to be
$$ \int_\Omega f(t) \, d\mu(t) = \lim_{n \to \infty} \int_\Omega f_n(t) \, d\mu(t).$$
Notice that the left integrals are easy to calculate for the simple functions in an obvious fashion. Finally, for $1 \leq p < \infty$, one can define $L^p$-spaces analogously to the scalar case as
$$ L^p(\Omega; X) := \{f : \Omega \to X : f \text{ is $\mu$-measurable and } \Vert f \Vert_p < \infty \}, \quad  \Vert f \Vert_p := \left( \int_\Omega \Vert f(t) \Vert^p \, d\mu(t) \right)^{1/p}.$$
Completely analogously to the scalar value case $L^p(\Omega; X)$ is a Banach space, the simple functions are dense in $L^p(\Omega; X)$ and convergent sequences in $L^p(\Omega; X)$ have dominated subsequences that converges almost everywhere in $X$. Finally, another nice feature is that the set
$$L^p(\Omega) \otimes X = \operatorname{span} \{f \otimes x : f \in L^p(\Omega), x \in X \}$$
is dense in $L^p(\Omega; X)$. Thus many things that hold in $L^p(\Omega)$ generalize to $L^p(\Omega; X)$. Since $X = \mathbb R^n$ with any norm is a Banach space it is just a special case of this general construction which can be generalized even further to locally convex spaces for example (instead of using Banach spaces). I hope that answers your question :)
A: Here is an elementary answer without Bochner integrals:
Usually one defines $$L^p(\Omega,\mathbb{R}^n):=\{f=(f_1,...f_n):\Omega\rightarrow \mathbb{R^n}: ~f_1,...f_n \in L^p(\Omega,\mathbb{R})\}$$
with norm $$\Vert f \Vert_p=({\Vert f_1\Vert_p^p+\dots +\Vert f_n\Vert_p^p})^{1/p}.$$
