Parameterized regular-zero counting for a full-rank linear family

Matrix inversion is $C^1$ on invertible matrices by its adjugate-over-determinant formula, so $a_0$ and $\mathrm{\Psi }$ are $C^1$. The entire incidence set is the graph $(z,N)=(z,\mathrm{\Psi }(z,b))$: solving $F=0$ is exactly solving $a=a_0(z,b)$. Differentiating $F(z,\mathrm{\Psi }(z,b))=0$ in $z$, with $b$ fixed, gives $A(z)D_z{a}_0=-D_{z}F(z,\mathrm{\Psi }(z,b))$. Thus $D\mathrm{\Psi }=\left(\begin{array}{cc}-A^{-1}{D}_{z}F& D_b{a}_0\\ 0& I_{k-d}\end{array}\right),\mathrm{\mid }\det D\mathrm{\Psi }\mathrm{\mid }=\mathrm{\mid }\det A{\mathrm{\mid }}^{-1}\mathrm{\mid }\det D_{z}F\mathrm{\mid }\mathrm{.}$ In particular an incidence point is critical for $\mathrm{\Psi }$ exactly when its root is not regular.

The open domain $U\times {\mathbb{R}}^{k-d}$ has dimension $k$, equal to the target dimension of $\mathrm{\Psi }$. By [F1], its critical-value set $E$ is Lebesgue null. If $N\mathrm{\notin }E$, no root can be irregular by step 1.1. This proves almost-everywhere regularity, including all roots in a noncompact $U$, not just a finite preselected root set.

Let $R$ be the open regular locus of $\mathrm{\Psi }$. Every point of $R$ lies in a rational open ball contained in a local inverse domain from [F2]. Take all rational balls on which $\mathrm{\Psi }$ is injective with invertible derivative; they form a countable cover $(V_j)$ of $R$ by [F5]. On each $V_j$, local inverses agree by injectivity, so $\mathrm{\Psi }{\mathrm{\mid }}_{V_j}$ is a $C^1$ diffeomorphism onto its open image. Put $E_j=V_j\setminus {\bigcup }_{i<j}{V}_i$. These are disjoint Borel sets covering $R$; their images under the corresponding diffeomorphisms are Borel by [F4]. No images of arbitrary noninjective Borel maps are asserted to be Borel.

Write ${\mathrm{\Psi }}_j^{-1}$ for the inverse on $\mathrm{\Psi }(V_j)$. The regular preimages of each $N$ are discrete, hence countable: each has a basis ball containing no other preimage and the first such rational ball gives an injection into a countable basis. Each regular root corresponds to exactly one point of $R$, which belongs to exactly one $E_j$. Therefore $C_{\phi }(N)$ is the sum over $j$ of the Borel functions equal to $\phi (z,N)$ at $(z,b)={\mathrm{\Psi }}_j^{-1}(N)$ for $N\in \mathrm{\Psi }(E_j)$ and zero elsewhere. This proves its Borel measurability. By step 2.1 it counts all roots for $N\mathrm{\notin }E$; any alternative extension on $E$ is Lebesgue measurable by [F4].

Apply [F2] on $V_j$ to the nonnegative Borel function obtained from $w(N)\phi (z({\mathrm{\Psi }}_j^{-1}N),N)$ and the indicator of $\mathrm{\Psi }(E_j)$. It gives ${\int }_{\mathrm{\Psi }(E_j)}w(N)\phi (z({\mathrm{\Psi }}_j^{-1}N),N)dN={\int }_{E_j}w(\mathrm{\Psi }(z,b))\phi (z,\mathrm{\Psi }(z,b))\mathrm{\mid }\det D\mathrm{\Psi }(z,b)\mathrm{\mid }dzdb\mathrm{.}$ Sum over $j$ by [F3]. Step 3.1 identifies the left side with $\int w{C}_{\phi }$. The right side is the integral on $R$; adding its critical complement changes nothing because $\mathrm{\mid }\det D\mathrm{\Psi }\mathrm{\mid }=0$ there and the other factors are finite. Substitute step 1.1 and use Tonelli [F3] for the nonnegative integrand to obtain exactly the displayed identity. All these are equalities of nonnegative integrals, so no finiteness assumption or limit-interchange theorem is needed.

For fixed $z$, the affine linear map $(a,b)\mapsto (y,b)=(A(z)a+B(z)b,b)$ is invertible with real determinant $\det A(z)$. Its change-of-variables formula gives $dN=\mathrm{\mid }\det A(z){\mathrm{\mid }}^{-1}dydb$. Taking the ordinary Dirac point mass at $y=0$ produces the inner integral of the Statement. For a second minor, write its full coordinates as $(y,b^{\mathrm{\prime }})$. The transition from $(y,b)$ has block matrix $\left(\begin{array}{cc}I& 0\\ C& D\end{array}\right)$; its absolute determinant is $\mathrm{\mid }\det D\mathrm{\mid }=\mathrm{\mid }\det A^{\mathrm{\prime }}\mathrm{\mid }\mathrm{/}\mathrm{\mid }\det A\mathrm{\mid }$, since the two full coordinate maps have absolute determinants $\mathrm{\mid }\det A^{\mathrm{\prime }}\mathrm{\mid }$ and $\mathrm{\mid }\det A\mathrm{\mid }$. On $y=0$ this gives $d{b}^{\mathrm{\prime }}\mathrm{/}\mathrm{\mid }\det A^{\mathrm{\prime }}\mathrm{\mid }=db\mathrm{/}\mathrm{\mid }\det A\mathrm{\mid }$. Thus the two weighted fibre integrals agree exactly. Thus this delta notation is a well-defined parameter-fibre measure, rather than an unjustified limit under the outer integral.

For a rank-$d$ $C^1$ matrix $C(z)$, the sets where a specified $d$-column minor is nonzero are open and cover $U$. Refine them by rational coordinate balls; this is a countable cover by [F5], with an invertible fixed minor on each member. The preceding construction applies on each member. Enumerate this open cover as $(O_j)$ and use the disjoint Borel pieces $S_j=O_j\setminus {\bigcup }_{i<j}{O}_i$. The indicators ${\chi }_j={\mathbf{1}}_{S_j}$ are bounded Borel weights and sum to one; no smooth-partition theorem is required. On each open $O_j$, replace $\phi$ by ${\chi }_j\phi$. Every root receives total weight ${\sum }_j{\chi }_j=1$; [F3] sums both sides exactly. The exceptional parameter sets have null union by [F5]. The minor independence in step 5.1 gives the same fibre integral on overlaps. Hence the local identities glue and prove all remaining assertions.

For a Lebesgue-measurable $S\subset U$, choose $S_0,Z$ as in [F6]. Apply the already proved Borel identity to $\phi (z,N)={\mathbf{1}}_Z(z)$ and $w=1$. Its right side is zero by Tonelli [F3], because the outer $z$-support is null; the Jacobian and all other factors are finite at each $(z,b)$, and an infinite inner integral on that null set also has zero outer integral. Thus $\int C_{{\mathbf{1}}_Z}(N)dN=0$. For every integer $j\ge 1$, monotonicity and the indicator formula [F6] give $j^{-1}{\lambda }_k(\{C_{{\mathbf{1}}_Z}\ge j^{-1}\})\le \int C_{{\mathbf{1}}_Z}=0$. Their countable union is $\{C_{{\mathbf{1}}_Z}>0\}$, so it is null. Outside this null parameter set there is no regular root in $Z$, and the root counts restricted to $S$ and $S_0$ agree for every bounded Borel weight. The $S_0$ count is Borel by step 3.1, and completeness in [F4] makes the $S$ count Lebesgue measurable. Their weighted parameter integrals agree; their right-hand integrals also agree, since ${\mathbf{1}}_S-{\mathbf{1}}_{S_0}$ is supported on the null set $Z$ and Tonelli applies to the two nonnegative integrands separately. With the exceptional set from step 2.1 removed as well, this proves the claimed completed-measure extension for all roots and finishes the proof.