Dirac-delta counting formula for regular zeros

Let $K=\mathrm{supp}\psi$, a compact subset of $U$. Every point of $f^{-1}(0)\cap K$ is isolated by [F1]. This zero set is finite: if it were infinite, compactness of $K$ would give a convergent subsequence of distinct zeros with limit $p\in K$; continuity gives $f(p)=0$, while the inverse-function neighborhood of $p$ from [F1] contains no other zero, a contradiction.

For each of these finitely many zeros $p$, choose a neighborhood $V_p⋐U$ on which $f$ is a diffeomorphism onto an open neighborhood $W_p\ni 0$. Shrink the neighborhoods to be pairwise disjoint. The compact set $K\setminus {\bigcup }_p{V}_p$ contains no zero, so continuity of $f$ gives a positive lower bound for $\mathrm{\parallel }f(z)\mathrm{\parallel }$ there. Choose $ϵ$ small enough that $B(0,ϵ)\subseteq W_p$ for every $p$ and that no point of $K\setminus {\bigcup }_p{V}_p$ maps into $B(0,ϵ)$. If there are no such zeros, compactness gives the same positive lower bound on all of $K$, and the integral is zero for sufficiently small $ϵ$.

It is enough first to take $\psi \ge 0$; a real $\psi$ is the difference of its nonnegative continuous compactly supported positive and negative parts. For $\psi \ge 0$ and $ϵ$ as in step 2.1, the integrand vanishes on $K\setminus {\bigcup }_p{V}_p$. Apply [F2], under the countable-choice hypothesis [F3], to $f{\mathrm{\mid }}_{V_p}:V_p\to W_p$ with the nonnegative measurable integrand $q(y)=\psi (f{\mathrm{\mid }}_{V_p}^{-1}(y)){\kappa }_{ϵ}(y)$. Change of variables makes the integral over $V_p$ equal to ${\int }_{W_p}\psi (f{\mathrm{\mid }}_{V_p}^{-1}(y)){\kappa }_{ϵ}(y)d^{d}y$. Summing these finitely many terms gives the original integral.

The function $y\mapsto \psi (f{\mathrm{\mid }}_{V_p}^{-1}(y))$ is continuous at $0$ and has value $\psi (p)$. Its integral against ${\kappa }_{ϵ}$ is the average over $B(0,ϵ)$; continuity implies that this average tends to $\psi (p)$. The sum is finite by step 1.1, so taking the limit in step 3.1 gives the claimed sum for nonnegative $\psi$, and subtraction gives it for every real $\psi$.

The limiting distribution assigns each regular zero $p$ the unit mass $\psi (p)$; therefore its density notation is exactly ${\delta }_0(f(z))\mathrm{\mid }\det Df(z)\mathrm{\mid }$, with no additional multiplicity or orientation sign.