Thank you for your explanation. The simple one is the one I was interested the most. I have heard arguments similar to your second explanation based on deeper results about Banach lattices and it was nice of you to present one such explanation too.

do you know if a simple proof that $L^+_p(\mu)$ has empty interior iff $L_p(\mu))$ has infinite dimension? or a reference perhaps?

@Asaf: It does not matter if the total mass $\mu(M)$ (volume element of the manifold at $M$) is infinity, as long as $\nu(M)=\mu(M)$ and possibly $\nu\ll\mu$, can a pushforward can be construct. The manifold can be assume compact and smooth to make things easier.

@Asaf: For example, the unit circle $\mathbb{S}^1$ in $\mathbb{R}^2$, its length element is the Hausdorff measure $H^1$ in $\mathbb{R}^2$ restricted to $\mathbb{S}^1$ (or a constant multiple of it); the surface area element of the unit sphere $\mathbb{S}^2$ in $\mathbb{R}^3$ is (a constant multiple) of $H^2$ in $\mathbb{R}^3$ restricted to $\mathbb{S}^2$.

I just posted a related question following your suggestion. Thanks!