Given a $\sigma$-algebra $\scr F$ on $\Omega$, say that an accuracy scoring rule for $\scr F$ is a function $s$ from the set of all (countably additive) probabilities on $\scr F$ to the $\scr F$-measurable functions on $\Omega$ with values in $[-\infty,M]$ (for some fixed real $M$). A scoring rule $s$ is proper iff $\int_\Omega s(p) \, dp \ge \int_\Omega s(q) \, dp$ for all pairs of probabilities $p$ and $q$, and strictly proper if additionally equality only holds when $p=q$.

If $\scr F$ is countably generated, it has a strictly proper scoring rule (see my answer here). The same is true for any measure space that has only atomic probability measures, like the one here.

Question:

1. Are there any strictly proper scoring rules for a $\scr F$ that is not countably generated and that has a nonatomic probability measure?

2. If yes, is this true for all $\scr F$?