On local convexity in $mathbb{L}^0$ and switching probability measures

In the paper, we investigate the following fundamental question. For a set $mathcal{K}$ in $mathbb{L}^0(mathbb{P})$, when does there exist an equivalent probability measure $mathbb{Q}$ such that $mathcal{K}$ is uniformly integrable in $mathbb{L}^1(mathbb{Q})$. Specifically, let $mathcal{K}$ be a convex bounded positive set in $mathbb{L}^1(mathbb{P})$. Kardaras [6] asked the following two questions: (1) If the relative $mathbb{L}^0(mathbb{P})$-topology is locally convex on $mathcal{K}$, does there exist $mathbb{Q}sim mathbb{P}$ such that the $mathbb{L}^0(mathbb{Q})$- and $mathbb{L}^1(mathbb{Q})$-topologies agree on ${mathcal{K}}$? (2) If $mathcal{K}$ is closed in the $mathbb{L}^0(mathbb{P})$-topology and there exists $mathbb{Q}sim mathbb{P}$ such that the $mathbb{L}^0(mathbb{Q})$- and $mathbb{L}^1(mathbb{Q})$-topologies agree on $mathcal{K}$, does there exist $mathbb{Q}sim mathbb{P}$ such that $mathcal{K}$ is $mathbb{Q}$-uniformly integrable? In the paper, we show that, no matter $mathcal{K}$ is positive or not, the first question has a negative answer in general and the second one has a positive answer. In addition to answering these questions, we establish probabilistic and topological characterizations of existence of $mathbb{Q}simmathbb{P}$ satisfying these desired properties. We also investigate the peculiar effects of $mathcal{K}$ being positive.