The simplest possible sampling of a function in the region [0,1] for the purpose of finding the area under the curve, for instance, could be just take the left offsets of the bars. That is, you have divided your [0,1] into n vertical bars and sample f(left border of every bar). If we take only 2 points, they will be 0 and .5. This has unsatisfactory shift to the left.
To fix it, you decide to sample in the centers of the bars. That is, you have the same 2 sample points, you take them in .25 and .75 -- right in the middles of your two bars. Looking at it, you immediately realize that it is not better -- you sample too closely to the 0 and 1 but leave a big gap in between .25 and .75. You realize that fair sampling would be to have 3 subintervals instead of two and you divide [0,1] into n+1 sections by exactly n points. Now, two points will sample at .33 and .66 -- equidistantly both from the interval borders as well as from each other. How do you call this kind of uniform sampling?
Carry me over to the math please if you think that deterministic sampling that I address here is not suitable in probability/statistic area.