Confusion about the Lebesgue spaces on Riemannian manifolds

Question

Having taken a few courses on measure theory and Riemannian geometry, I still fail to make a successful access to the Lebesgue space $L^p(M)$ on a Riemannian manifold $(M,g)$. In the 1987 article THE YAMABE PROBLEM authored by Lee and Parker, $L^p(M)$ is defined to be the set of locally integrable functions $u$ on $M$ for which the norm $$\lVert u\rVert_p=\left(\int_M|u|^p dV_g\right)^\frac{1}{p}$$ is finite. This definition is not for the authors' exclusive use and can be commonly found in the literature, but the motivation to consider the integral $$\int_M|u|^p dV_g$$ is totally mysterious to me. According to Measure and Integral: An Introduction to Real Analysis by Wheeden and Zygmund, constructing Lebesgue spaces requires a measure space, which is not a problem in our present case because the Riemannian metric $g$ induces a distance function on $M$ and hence helps us build an outer measure. Then it would be natural to define integrals by using this very outer measure. If this is the case, why would I have to consider the integral $$\int_M|u|^pdV_g$$ using the Riemannian volume form, not to mention that $|u|^p$ may not be compactly supported? Can someone give me an idea of what's going on in the formation of $L^p(M)$? Thanks for everything.

Update: Thank you all, and I was astounded to know that if one is to have a well-defined distance function on $M$, then $M$ has to be connected in order that any two points of $M$ can be joined by a piecewise smooth curve segment. Maybe that's why I should refrain myself from using the outer measure induced by the Riemannian distance function.

Answer 1

Let $(M,g)$ be a pseudo-Riemannian manifold. The metric tensor $g$ gives rise to a unique positive measure $\lambda_g$ defined on the Lebesgue $\sigma$-algebra, $\mathscr{L}(M)$, of $M$ such that it has the desired formula in terms of a chart (see the link for details). It is also common to denote this measure as $V_g$ or $\mu_g$ (or $dV_g$ or $d\mu_g$, or even $dA_g$ if you’re talking about submanifolds of codimension one, or one might even suppress $g$ in the notation).

Now we have a measure space $(M, \mathscr{L}(M), \lambda_g)$, in the strict real-analysis measure-theory sense, so all that theory carries over verbatim. In particular, for $p\in [1,\infty]$, we can define the Lebesgue spaces $L^p(M)\equiv L^p(\lambda_g)\equiv L^p(M,\mathscr{L}(M),\lambda_g)$, which are Banach spaces as usual.

A remark about notation: we often overload the notation so that $dV_g$ denotes the measure induced by $g$, or alternatively on an oriented manifold it is also used to denote the induced volume $n$-form. In the oriented case the integral of an $L^1$ function $u$ with respect to the measure $dV_g$ is equal to the integral of the $L^1$ differential$n$-form $u\,dV_g$ (if you’re worried about integrating differential forms of non-compact support, take a look at this answer). The fact that these are equal follows immediately by unwinding definitions (of the measure and of the volume $n$-form).