Setup: Let $(M,g)$ be a (possibly non-compact) Riemannian manifold with volume density $d_gV$. Then one may think of $(M,g)$ as a measure space $(\Omega,\mathcal{A},\mu)$, where $\Omega:=M$, $\mathcal{A}:=\sigma(\tau_M)$ is the $\sigma$-Algebra generated by the topology $\tau_M$ of $M$ and for any $A \in \mathcal{A}$, $\mu(A):=\int_M{\chi_A d_gV}$, where $\chi_A:M \to [0,1]$ is the characteristic function of $A$. We obtain $\int_{M}{f d\mu} = \int_M{f d_gV}$, where the left hand side is understood to be an integral in the measure theoretic sense and the right hand side is an integration of a density. This enables us to define the space $L_p(\mu)$ with norm $\|f\|_{L_p(M)}^p = \int_{M}{|f|d_gV}$ on a manifold and apply all the results from integration theory to it, e.g. that it is a Banach space and so on.
My question is: Does this work in the following more general setup: Extend the Riemannian metric on $M$ to a fibre metric in $\bigwedge^k T^{\;*}M$, $0 \leq k \leq m$, (as described in the paragraph below). Then one may define $L_p$-spaces of differential forms by setting $\|\omega\|_{L_p(M)}^p := \int_{M}{|\omega|^pd_gV}$ and setting $L_p^k(M)$ to be the space of all measurable $k$-forms on $M$ (i.e. with Lebesgue measurable coefficient functions in any chart) such that $\|\omega\|_{L_p(M)}<\infty$. Is it possible to construct a measure space $(M,\mathcal{A},\mu)$ such that $L_p^k(M)$ may be thought of as an $L_p(\mu)$ as well?. The problem obviously is the range of a differential form. Formally it is a map
$\omega\colon M \to \bigwedge^k T^*M$, i.e. it takes values in the vector bundle $\bigwedge^kT^*M$. Even if integration theory is available for functions on measure spaces with values in Banach spaces, this does not help since the bundle itself is not a vector space. I am interested in this question, because otherwise I see no alternative but to establish all the results about integration theory for $L_p^k(M)$ again, i.e. that it is a Banach space, Lebesgue Dominated Convergence Theorem, Fubini/Tonelli etc. That seems a bit exaggerated since intuitively this space is not so fundamentally different.
Construction of the fibre metric: For any $0 \leq k \leq m$ the Riemannian metric may be extended canonically to differential forms in $\Omega^k(M)$ in the following way: For one forms $\omega,\eta \in \Omega^1(M)$ define $g(\omega,\eta):=g(\omega^\sharp, \eta^\sharp)$, where $\sharp:T^*M \to TM$ is the sharp operator with respect to $g$. Then define $g$ on decomposable forms by $g(\omega^1 \wedge \ldots \wedge \omega^k, \eta^1 \wedge \ldots \wedge \eta^k):= \det(g(\omega^i, \eta^j))$.
Best Answer
Let $E$ denote a vector bundle over a manifold $M$ equipped with a metric, and $L_p(E)$ the space of measurable sections of $E$ with finite $L_p$ norm. Obviously, in general, one can't identify $L_p(E)$ with an $L_p$ space of vector valued functions.
First assume that $M$ is compact. To understand $L_p(E)$, we use a finite set of trivializations $(U_i, h_i)$ of $E$ which cover $M$. Each trivialization identifies $E|_{U_i}$ with $U_i \times\mathbf R^n$ (or $U_i \times\mathbf C^n$). We choose the trivializations such that the bundle norm is equivalent to the euclidean norm, i.e. bounded from above and below. Then an $L_p$ section of $E$ is equivalent to a set of $\mathbf R^n$ (or $\mathbf C^n$)-valued measurable functions ${f_i}$ on $U_i$ satisfying the transition law, such that $\sum \|f_i\|_p$ is finite.
Using this one can easily extend all the basic theorems to $L_p(E)$. In particular, one shows that $L_p(E)$ is equivalent to the completion of $C^{\infty}(E)$ (the space of smooth sections on $M$) w.r.t. the $L_p$ norm.
If $M$ is noncompact, we write it as the union of locally finite compact subsets $A_i$, such that the intersections of $A_i$ have zero measure. Then the $L_p$ norm of a section $s$ is given by $(\sum \int_{A_i} |s|^p)^{1/p}$. Then the arguments for the compact case can easily be carried over. (We argue on each $A_i$, and then combine.)