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Larks comments on Infinitesimals: Another argument against actual infinite sets - Less Wrong
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Comments (31)
No, the probability density function for a uniform distribution on [0,1] is what you are integrating, and that is non-zero.
Is it? How probable is 1/2, for example?
That's not what a probability density function is.
Still. How probable is 1/2 in the above process of coin toss?
1/2=.1000000... in the binary presentation, means one head and all tails.
<nitpick> Or also .0111..., one tail and all heads. </nitpick>
Any individual number has probability 0, but the probability density is the probability that you'll get a number between x and dx, divided by dx, in the limit as dx approaches 0.
Any individual real number has the zero probability, but at least one of them - is bound to happen.
One may or may not consider sub intervals. It is a side question. Just as rational numbers, or algebraic numbers on this interval. Every sub-interval has the probability equal of its length what is always nonzero. All rational numbers have the probability 0, for example.