[l33t2328]

You’re certainly aware of this, but for those that aren’t: asking that the “probability” in the space to be well defined you are essentially invoking the idea that the function is _measurable_. Measure is a way to generalize things like length and volume to sets which have no length or volume in the traditional sense. One good way to do this is say that lines have “measure” equal to their length, and if you can combine line segments(even infinitely many) to get some new line then that line has measure as well. If you do make this precise, you get the so-called “Borel measure” on the real line.

If you want all subsets of sets with 0 Borel measure to also have 0 measure, then this leads to the notion of the lebesgue measure, and it can be used to define the lebesgue integral.

[pfortuny]

Two insights more

a) But for probability you nees the measure of the whole space to be 1, which forces points “far away” to have very small “weight”. Thus, there is no uniform distribution in the whole R.

b) and then your mind blows up when you realize that discrete probability is the very same thing, and that an integral in a finite set is just summation.