The author says in a comment

> If hitting exactly zero is much less probable than hitting exactly 0.5, then I would argue that you do not have a uniform distribution…

But due to the non-uniformity of floating point numbers, I think the author is wrong here.

We can think of each floating point number as an interval. The probability that it is picked should ideally be equal to the size of the interval.

Yes. The effect of having a coarser partition around a mesh point is countered by the effect of a higher frequency of hitting that partition. This actually is the essence of importance sampling. For the purposes of numerical integration, as long as the integrand is relatively smooth with respect to the partition size, there shouldn't raise any serious problem. On the contrary, if the integrand does vary wildly between two adjacent floating point numbers, then the real issue here is that the precision (single, double, quadruple, etc.) used is not fine enough, rather than the deviation from uniform distribution.

Exactly.