No, propositions are simply statements for which is it meaningful to assign a truth value. It might be possible to come up with a proposition that is associated with a non-measurable set, though I can't offhand think of an example. But remember: Bayesian probabilities are models of belief. Priors are peronsal. And so in order to assign a prior to a proposition, the statement of the proposition must have some referent in your personal ontology. You can't hold a belief about the truth value of a statement unless you know (or at least think you know) what that statement means. So unless a person has something in their ontology that corresponds to a non-measurable set (and I suspect most people don't) then that person cannot assign a Bayesian prior to a statement associated with a non-measurable set. But that's a limitation of that particular individual, not a limitation of Bayesian reasoning. For example: you cannot assign a Bayesian prior to the statement, "The frobnostication of any integer is even" because you don't know what a frobnostication is.

(Note that there are all kinds of ways that statements can fail to be propositions. For example, you can't assign a Bayesian prior to the statement: "All even integers are green" despite the fact that you know what all the words mean.)