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](https://mathoverflow.net/questions/374094/example-of-a-strictly-proper-scoring-rule-on-a-general-measurable-space)). The same is true for any measure space that has only atomic probability measures, like the one [here](https://www.ams.org/proc/1983-089-04/S0002-9939-1983-0718982-7/S0002-9939-1983-0718982-7.pdf).


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 $\scr F$?


**Note:** The answer to (2) is negative if there are cases where there are more than ${\mathfrak c}^{|\Omega|}$ probability measures on $\scr F$. By a result of [Paris and Koonen](https://www.sciencedirect.com/science/article/pii/0003484371900015), it is relatively consistent with the existence of a measurable cardinal that there be a measurable cardinal $\kappa$ that has $2^{2^\kappa}$ normal measures, so it is relatively consistent with the existence of a measurable cardinal that the answer to (2) is negative.



Existence of a strictly proper scoring rule on a $\sigma$-algebra that is not countably generated
There is a somewhat boring positive answer to the first question. It is shown in [K. P. S. Bhaskara Rao, and B. V. Rao. [Borel spaces.](https://eudml.org/doc/268562) Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1981.] on page 15 that the $\sigma$-algebra on $[0,1]$ generated by analytic sets is not countably generated. However, since analytic sets are universally measurable, every Borel probability measure has a unique extension to this $\sigma$-algebra. It follows that every proper scoring rule for the probability measures on $[0,1]$ with the Borel sets works also for this larger space that is not countably generated.


