Suppose that $\Omega$ is an open subset of $\mathbb R^n$.
Let $p \in (0,\infty)$. We say that $f : \Omega \rightarrow \mathbb R$ is in the space $L^p(\Omega)$ if it satisfies $$\int_\Omega |f|^p dx < \infty.$$
These spaces are quasi-Banach spaces, Banach spaces for $p \geq 1$, and there are known bounded inclusions when $p \geq 1$ and the set $\Omega$ has finite measure: $$ L^q(\Omega) \subseteq L^p(\Omega) \text{ if } q > p. $$ Question: What inclusions are known over the entire range $p > 0$ (under what conditions on $\Omega)$?
I suspect that $L^1(\Omega) \subseteq L^p(\Omega)$ as an inclusion of sets whenever $\Omega$ is bounded and $p < 1$, but is that embedding a bounded inclusion?