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.
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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.
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You can't have a uniform distribution on the reals, so it wouldn't be a probability space. Any pdf needs to be in L^1
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Basically, if you have a pdf for real numbers you are randomly choosing, with full support, that are truly unbounded, then any open interval has positive probability.
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That kind of fits with my ides of this as a limiting process....the probability may be arbitrarily small...but it's not truely zero.
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