As we know, the delta function is not a Radon-Nikodym density with respect to the Lebesgue measure.
If we choose the counting measure $\mu$, which assigns to every set the number of its elements, then the delta function should be a density.
But how do we compute this integral then: $$\int \delta(x) d\mu(x) = \int_{\{x=0\}} \delta(x) d\mu(x)$$
How can this be evaluated to make sense?