To add a slightly different angle to [PhotonicBoom's sound answer](http://physics.stackexchange.com/a/141214/26076), the link between the two entities -  discrete sum and integral - is the concept of *measure*, not of *limit*. You can think of your sum as a Lebesgue integral if you choose a [discrete measure](http://en.wikipedia.org/wiki/Discrete_measure) for the real line with the measure's "anchors" at a countable set of "allowed values". Discrete and continuous measures are highly analogous insofar that they both have all the "Real MacCoy" properties of measures: non-negativity, positivity and countable ($\sigma$-) additivity. Ultimately, though, the two are as different as are $\aleph_0$ and $\aleph_1$, dramatically illustrated by the Cantor Slash argument: a quantum observable which can in principle yield any real number in an interval as a measurement and one which can only have discrete values as measurements are very different beasts.

In quantum mechanics, or at least all the QM I've seen, one makes an assumption of a *separable* or a *first countable* Hilbert space for the state space. This means, in effect, that there exists a countable basis for the space of states: for example, the quantum harmonic oscillator's state can be expressed as a superposition of the countable set of energy eigenstates. So in "normal QM", there is always a co-ordinate transformation which will turn an integral completeness relation into a discrete one, although, at the same time, you are changing the observable whose eigenstates span the state space.