When you talk about a measure on an infinite set, you have already made the "hidden limit" step without noticing. There is no empirical evidence for the real existence of anything like "infinite set", nor is it conceivable what sort of evidence would be required. Infinite sets, and continuous...
Only if you use sloppy language. The non-sloppy and non-confusing statement is that probability tends towards zero in the limit as you reduce the interval's size towards 0. Furthermore, it seems there are no continuous variables in the real world; for all practical purposes actual phenomena are...
Wrong.
Let me explain probability to you from a Bayesian perspective. It may be more helpful than what you're taught at school. The gist of it is translating expressions like P(A|B) into English (it reads "probability of A given B" - but what does it mean - read on!).
A - some well-defined...