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 …
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Yet I potentially could pick a natural number...which tells me that the probability while very small is not zero (even arbitrarily small)
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The Strong Law Of Large Numbers provides a nice example of this “almost surely” idea https://en.wikipedia.org/wiki/Law_of_large_numbers#Strong_law …
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Is the probability of choosing a number in the real line the same as choosing a number in real half line? Both zero? It seems a paradox to me because you have fewer options, so the probability should increase...
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What is a real half line? Is it the real numbers bigger than some fixed one? If so, that's isomorphic to the real line.
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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.
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Right, but even if you chose a small interval of reals, the logic is the same
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