Let $X$ be a measurable space and let $\mu$ be a probability measure on $X$ (assumed to be non-atomic, in case that helps). Let $v:X \to \mathbb R^k$ be a $\mu$-integrable function and for every $t \in [0,b]$ (with $b>0$ fixed), consider the function $G_t:\mathbb R^k \to [0,1]$ defined by $$ G_t(w):= \mu(\{x \in X \mid w^\top v(x) > t\}). $$
Question. Under what minimal conditions on $\mu$ and $v$ are the functions $G_t$ continuous for (almost) all $t \in [0,b]$ ?
