〈 Berger | Dillon 〉

As a physicist I have issues with this. If an event has occurred within any arbitrary time period than it's probability of occurring must not have been zero...but rather some epsilon. Maybe this is a limit thing? I'll have to look into it.

Think of it like this. If I ask you to pick a number on the real line, they you *almost surely* wouldn't pick a natural number, since they have zero measure in the reals. https://en.wikipedia.org/wiki/Almost_surely …

In real life you can get around that because your observation is never a specific real number, but rather some small interval of real numbers.

Right, but even if you chose a small interval of reals, the logic is the same

Is it? A small interval has non zero measure. You have to assume that a real value for the random variable on the continuum exists. Maybe all you ever get are intervals of various sizes

This is what I mean (it may not be what you intended to say): Consider a segment of the real line w. length very large N. If you place an ε interval around each integer, then the P that you pick the interval around 7 is 2ε/N, but as N->∞ this probability vanishes.