So if I ask you if you think I can hit immeasurable set A on a dartboard, you'd say 50%. Same with disjoint immeasurable set B. Same with A U B. I now offer to bet with 1:1 odds that you can hit A, can hit B and can't hit A U B. If you hit A, you win the first bet, but lose the second two. If you hit B, you win the second, but lose the other two. If you miss both, you lose all three. No matter what, I get money.
Hm. The simplest way around this is to treat the fact that an immeasurable disjoint set B exists as new information to our agent. E.g. if you just tell me to bet on hitting some immeasurable set A, I'll think the possibilities are just (A) or (not A), and in my state of ignorance will bet at 1:1 odds. But if you then tell me there's some disjoint set B, now the possibilities are (A), (B), (neither). Maximum entropy dictates that I only assign a 1/3 probability to hitting A or B. This handles the dutch book correctly.
If you add knowledge about relation...
I got into a heated debate a couple days ago with some of my (math grad student) colleagues about the Sleeping Beauty Problem. Out of this discussion came the following thought experiment:
Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: She will be put to sleep. During the experiment, Beauty will be wakened, interviewed, and put back to sleep with an amnesia-inducing anti-aging drug that makes her forget that awakening. A fair coin will be tossed until it comes up heads to determine which experimental procedure to undertake: if the coin takes n flips to come up heads, Beauty will be wakened and interviewed exactly 3^n times. Any time Sleeping Beauty is wakened and interviewed, she is asked, "What is your subjective probability now that the coin was flipped an even number of times?"
I will defer my analysis to the comments.