Let $\mu$ be a finite measure on the measure space $(\mathbb{R}^d,\Sigma)$, $K\subset \mathbb{R}$ be compact of positive $\mu$-measure, and defined $\frac{d \nu}{d\mu}\triangleq 1_K$ as well as the modified $L^p$ space $$ \mathbb{L}^p_{\nu}(\Sigma)\triangleq \left\{ f \in L_{\mu,loc}^p(\Sigma): \, \int_{\mathbb{R}^d} |f|^p d\nu <\infty \right\} \subseteq L^p_{\nu}(\Sigma). $$
It is easy to see that in general, the inclusion is strict.
Besides the above inclusion, how are the spaces $\mathbb{L}^p_{\nu}(\Sigma)$ and $L^p_{\nu}(\Sigma)$ related? Is the former dense in the latter? If not, is it a Banach space?
My intuition: $\mathbb{L}_{\nu}^p(\Sigma)$ is a dense subspace of $L_{\nu}^p(\Sigma)$ and it is a locally-convex (TVS) but not Banach.