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

Abstract

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 ${\mathbb{Q}}^{\prime }\sim \mathbb{P}$ such that $\mathcal{K}$ is ${\mathbb{Q}}^{\prime }$-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 $\mathbb{Q}\sim \mathbb{P}$ satisfying these desired properties. We also investigate the peculiar effects of $\mathcal{K}$ being positive.