Let $(\Omega, \mathcal{B})$ be a measurable space (we can assume that $\Omega$ is metric, compact, etc. if it is helpful) and let $u$ and $v$ be probability measures on $\Omega$. Denote by $\mathcal{L}(u)$ the space of all $u$-integrable functions.
What are some general conditions on $u$ and $v$ underwhich $\mathcal{L}(u) \subseteq \mathcal{L}(v)$? (a complete characterization would be ideal!)
For example, a relatively trivial condition would be: $v$ is absolutely continuous with respect to $u$ and $dv/du$ is bounded.