On a Riemannian manifold $(M,g)$, let $\mathcal L^p(M,k)$ denote the space of measurable $k$-tensors $T$ (i.e., the coordinate components in any chart are Lebesgue measurable) for which the norm $$||T||_p=\int_M |T|^p\,d\mu_g,\quad |T|=\sqrt{T_{i_1\cdots i_k}T^{i_1\cdots i_k} }$$ is finite. Is this space complete?

In compact case, I obtained an affirmative answer on Math Stack Exchange, but the answer does not seem to carry over to the general case. The book by Chavel, *Eigenvalues in Riemannian Geometry*, seems to claim this is true. He offers no proof.