I’ve read several answers about how how crazy sets like Vitali’s set are (one of the reasons) why we need $\sigma$-algebra and measure theory for probability spaces. However, Vitali’s set is only eliminated by requiring non-trivial translation-invariant measures. So either (a) Vitali’s set wasn’t a problem in the first place or (b) probability spaces need translation-invariance.
No definition of probability spaces I’ve seen mentions translation invariance, leaving us with (a): Vitali’s set is not a problem for probabilists. Then:
- Why do people mention Vitali's set while explaining the measure-theoretic formulation of probability?
- Why do, in fact, we need measure theory for probability?
- Are there any other paradoxes arising from uncountable sets to worry about that do apply to regular measures and probability?