Integration of differential forms using measure theory?

Question

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 $g(\omega^1 \wedge \ldots \wedge \omega^k, \eta^1 \wedge \ldots \wedge \eta^k):= \det(g(\omega^i, \eta^j))$.

Answer 1

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.)

Answer 2

Yes, L_p-spaces can be defined for arbitrary hermitian vector bundles.

For the sake of convenience I denote L_p=L^1/p (see this answer for a motivation), in particular L₀=L^∞ and L_1/2=L². As explained in the link above, p is an arbitrary complex number such that ℜp≥0.

Suppose M is an arbitrary smooth manifold, possibly non-compact, and V is a finite-dimensional hermitian vector bundle over M. Let me stress that we do not need any additional data on M such as a metric, a volume form, a density, or an orientation.

Recall the definition of the line bundle Dens_p(M) of p-densities on M for an arbitrary complex number p (no restrictions on the real part of p): Every fiber of Dens_p(M) is the vector space of all set-theoretical maps f: Λ^top(TM) \ {0} → C such that for all λ∈C \ {0} and for all x∈Λ^top(TM) \ {0} we have f(λx)=|λ|^pf(x). In particular, Dens₁(M) is the tensor product of the line bundle of top-degree differential forms on M and the line bundle of orientations of M.

Note that for all p∈R the line bundle Dens_p(M) has a canonical orientation, in particular it is trivaliazable. Moreover, for all p∈C \ R the line bundle Dens_p(M) is also trivializable, even though it does not possess a canonical orientation. However, only Dens₀(M) has a canonical trivialization.

Observe that all bundles Dens_p combine together in a C-graded unital *-algebra, i.e., we have the unit C→Dens₀(M), the multiplication Dens_p(M)⊗Dens_q(M)→Dens_p+q(M), and the involution (Dens_p)*→Dens_p* (the first star denotes the conjugation of the complex structure on a vector bundle, the second star denotes the conjugation of complex numbers). All these morphisms are isomorphisms of line bundles.

For any t∈C and any p>0 we also have the power operation Dens_p⁺(M)→Dens_tp(M). This is not a morphism of vector bundles, because it is non-linear for t≠1 and Dens_p⁺(M) is a fiber bundle, not a vector bundle. However, the power operation is still a morphism of fiber bundles, in particular we can talk about powers of positive sections of Dens_p(M).

We have a canonical integration map ∫: C^∞(Dens₁(M))→C. We use this map to define norms on spaces C^∞(Dens_p(M)) for all p∈C such that ℜp>0. First we send f∈C^∞(Dens_p(M)) to |f|=(f*f)^1/2∈C^∞(Dens_ℜp(M)). Observe that f*f and |f| are positive with respect to the canonical orientations on Dens_2ℜp(M) and Dens_ℜp(M). Then |f|^1/ℜp∈C^∞(Dens₁⁺(M)) and we set ‖f‖:=∫⁠(|f|^1/ℜp). This is a norm for ℜp≤1 and a quasi-norm for ℜp>1.

The (quasi-)Banach space L_p(M) is the completion of C^∞(Dens_p(M)) in this (quasi-)norm.

If ℜp=0, then we complete C^∞(Dens_p(M)) in the weak topology induced by C^∞(Dens_1−p(M)) and obtain the Banach space L_p(M). (The norm of f∈C^∞(Dens_p(M)) can be defined as the supremum of |f|∈C^∞(Dens₀(M))=C^∞(M), however, C^∞(Dens_p(M)) is not dense in L_p(M) in the norm topology.)

Thus we defined L_p-spaces of the trivial line bundle on M for an arbitrary p∈C such that ℜp≥0. To extend this definition to an arbitrary hermitian vector bundle V we replace f*f by μ(f*f) in the above definition of norm. Here μ denotes the hermitan pairing on V.

All the usual theorems of measure theory like Radon-Nikodym, Riesz, Fubini, Tonelli etc. hold in this more general setting.

Answer 3

Thanks for your posts. I am summing up what we have got so far.

@Dmitri Pavlov: You explained why the answer to my question is negative and give an alternative approach to $L_p$-spaces on hermitian vector bundles in your answer. You claim that all the usual theorems of measure theory hold in this more general setting. Can you give a reference for that?

@Deane Yang: You claim that this can be done locally. That idea seems natural to me, but I'm afraid I cant make this rigorous: Let $E \to M$ be a real vector bundle of rank $k$ with fibre metric $h$ and let $(M,g)$ be a Riemannian $m$-manifold. Assume $U \subset M$ is open and sufficiently small such that there exists a chart $\varphi:U \to R^m$ and a local trivialization $\Psi=(\Psi_1,\Psi_2):\pi^{-1}(U) \to U \times R^k$. Let $\mu_g$ be the Riemannian volume density on $M$, $\tau_g:=\sqrt{\det(g_{ij})}$ and $\left\| \cdot \right\|_h$ be the norm induced by $h$. Then for any section $s \in \Gamma(E)$, we obtain

$$\int_{U}{\left\| s \right\|_h^p \mu_g} = \int_{V}{(\left\| s \right\|_h^p \tau_g \circ \varphi^{-1} )(x) dx}=\int_{V}{\left\| s(\varphi^{-1}(x)) \right\|^p_{h(\varphi^{-1}(x))} \tau_g(\varphi^{-1}(x)) dx }$$

Now of course $\tilde s:= \Psi_2 \circ s\circ \varphi^{-1}:V \to R^k$ is a vector valued function, but you can't choose a norm $\left| \cdot \right|$ on $R^k$ such that for any $x \in V$, we obtain $\left| \tilde s(x) \right| = \left\| s(\varphi^{-1}(x) \right\|_{h(\varphi^{-1}(x))} $, because the fibre metric may change in every point.