Understanding exactly what degrees of belief are, becomes a lot easier (I suppose) if you know that for some reason they are isomorphic to frequencies

If probabilities were systematically wrong about the frequency of success in independent trials, there would be some other method of reasoning from incomplete information that was better than probabilistic logic. But since the real world obeys all the requirements for probabilistic logic (basically, causality works), there is no such method, and so frequencies match probabilities.

for some reason they are also isomorphic to spatial measures

Read a introductory chapter on set theory that uses pictures to represent sets, and you will understand why.

It's certainly an interesting fact that these things behave the same. But it's not an unsolved problem. We don't have to keep a definition around that's useless in the real world because of any lurking mystery.