What mean $\int \mathcal D(n(x)) \delta \left(n(x)-\frac{1}{N}\sum_{k=1}^N \delta (x-x_i)\right)$?

Question

In formula (4.5) page 27 of this document, what mean $$\int \mathcal D(n(x)) \delta \left(n(x)-\frac{1}{N}\sum_{k=1}^N \delta (x-x_i)\right),$$ where the integral run over all possible normalized, non negative smooth function $n(x)$, and $\delta $ is the $\delta $ distribution. Could someone explain what is this integral, how it work and why this integral is $1$ ?

Answer 1

Looks like a path integral. Actually the result should be $\frac{1}{N}\sum_{k=1}^N \delta (x-x_i)$. Then when you integrate over $x$ one more time it is $=1$.

Look into distributions to get a explanation for the $\delta()$. The one appearing in your integral is a sort of infinite dimensional delta distribution.