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Thomas comments on Infinitesimals: Another argument against actual infinite sets - Less Wrong
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It is allowed in some systems, to pick a random real number from the interval [0,1]. The probability for the each of them is 0, yet the probability for the whole interval is 1.
A way to pick a random number from this interval is tossing a fair coin countably many times. The head gives you 1 and the tail gives you 0 in the binary representation. Every toss takes half the time as the previous one, so you finish this construction in a finite time. So called supertasks are allowed sometimes.
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.
The coin flipping trick will miss plenty of numbers, like one third - and those that are left have a small but non-infenitesimal probability.
Edit: whoops, my bad, read "countable" as "finite".
in the binary representation