I believe you've defined an equivalent if unusual form (or rather, your definition can be extended to an equivalent form).

Yeah, that's what MrMind said too. Thanks!

The only laws of probability measure I know are that the measure of the whole set is 1, and the measure of a union of disjoint subsets is the sum of their measures. I'm finding it hard to imagine how I could hold beliefs that wouldn't conform to them. I mean, I guess it's conceivable that I could believe that A has probability 0.1, and B has probability 0.1, and A OR B has probability 0.3, but that just seems crazy.

Yeah, and I fully grasp the "measure of the whole set is 1" thing. (After all, if you're 100% certain something is true, then that's the only thing you think is possible). The additivity axiom is harder for me to grasp, though. It seems like it should be true intuitively, but teaching myself the formal form has been more difficult. Thinking and Deciding tries to derive it from having different bets depending on how things are worded (for example, on whether a coin comes up heads or tails versus whether the sun is up and the coin comes up heads or tails) which I grasp intellectually, but I'm having a hard time grokking it intuitively.

I think you're trying to be too formal too fast (or else your title isn't what you're really interested in). Try getting a solid practical handle on Bayes in finite contexts before worrying about extending it to infinite possibilities and the real world.

I do have a subjective feeling of success when I use Bayes (or Bayes-derive heuristics, more commonly) in my everyday life, but I really want to be sure I understand the nitty-gritty of it. Even if most of my use of it is just in justifying heuristics, I still want to be sure that I can formulate and apply them properly, you know?