So this is basically from an economics lecture (the lecturer introduced the voting model from myerson 1993).
I have this feeling that the lecturer simplified this model though, so this might not be too related. Anyway, the idea is that you have a continuum of voters from the $[0,1]$ interval, and parties pick a redistribution platform. This is a distribution $t$ on the interval $[0,1]$ such that the expected value is $0$. Or in other words, the redistribution is zero sum.
Now the result is apparently, that picking the transfer $t(x)$ for individuum $x$ randomly (iid) out of $U([-1,1])$ for all $x \in [0,1]$ is a Nash equilibrium.
But is such a mapping $t:[0,1]\rightarrow[-1,1]$ even measurable?
We will only get a probabilistic answer of course, but my guess would be that this mapping would almost surely be not measurable, given that the number of measurable mappings is of the size of the continuum still, while the total number of mappings would be the powerset of the continuum if I remember correctly. And since your randomly pick function values you will probably end up with a function from the larger set. But this is only my intuition and I don't really know how to formulate this.
And does a continuum of iid random variables even exist?
