part 2

For idealized counters, imagined unreachable mathematical limit of a set of stepwise refined realizable counters, practical work becomes much easier and less clumsy, integrals instead of sums, functions instead of discrete objects. Kepp in the backyard though, that the mathematic subtleties are a high price, -often kept under the rug. A student can circumvent these, whenever in doubt, by replacing continuous states by discrete counters of a given length and then calculate the result, only then to do the limit to infinite smallness of the counters. The sets of stepwise refined couunters and their limit are

$$\vert x\!) = \underset{n \rightarrow 0}{\lim} \vert x, \Delta_n x\!> \quad \mathrm{with} \quad \Delta_n = \Delta/n \quad .$$ (5)