Finite sums of product tests are dense on product open sets

If $\mathrm{\Phi }=0$ use the constant sequence of empty sums. Otherwise let $A,B$ be the compact coordinate projections of its support. Take nonnegative smooth compact bumps in the two coordinate spaces from F1 and divide each by its positive finite Riemann integral. Denote them by $\rho ,\sigma$; positivity follows since each is one on a ball, and finiteness from bounded compact support. Fix radii $R_{\rho },R_{\sigma }$ containing their supports. Choose ${\epsilon }_0>0$ such that $A+\bar{B}(0,{\epsilon }_0{R}_{\rho })\subseteq U$ and $B+\bar{B}(0,{\epsilon }_0{R}_{\sigma })\subseteq V$, possible by compact interior margins. These two compact neighborhoods form the fixed support product $L$.

For $0<\epsilon \le {\epsilon }_0$ form the Riemann integral [step 1.1, F2, F3, F4, F5] $I_{\epsilon }(x,y)=\int \mathrm{\Phi }(a,b){\rho }_{\epsilon }(x-a){\sigma }_{\epsilon }(y-b)dadb\mathrm{.}$ A fixed box containing $\mathrm{supp}\mathrm{\Phi }$ bounds the parameter integral. On any fixed compact target box, take uniform parameter grids with midpoint tags. For every mixed target derivative $D^{\gamma }$, differentiating the corresponding finite sums gives the tagged sums for the $D^{\gamma }$-derivative of the integrand. Joint uniform continuity on the compact parameter-target product makes these sums converge uniformly in $(x,y)$: their error from the derivative-integral candidate is at most the parameter-box volume times the largest oscillation on a grid cell. The tagged-sum theorem in F5 identifies the pointwise candidate with the Riemann integral, and repeated applications of F3 on coordinate intervals identify it with $D^{\gamma }{I}_{\epsilon }$. Thus $I_{\epsilon }$ is smooth and supported in $L$.

The product kernel has Riemann integral one by F5, is nonnegative and has fixed bounded support. Uniform continuity of the globally smooth compactly supported $D^{\gamma }\mathrm{\Phi }$ therefore bounds $\sup \mathrm{\mid }{D}^{\gamma }{I}_{\epsilon }-D^{\gamma }\mathrm{\Phi }\mathrm{\mid }$ by its modulus of continuity at $\epsilon \sqrt{R_{\rho }^2+R_{\sigma }^2}$, tending to zero. Uniform continuity on all space follows from uniform continuity on a compact neighborhood of its support and vanishing outside it. This proves convergence of every derivative as $\epsilon \downarrow 0$.

Put $P_0=0$. For $j\ge 1$, set ${\epsilon }_j={\epsilon }_0\mathrm{/}(j+1)$ and, for the original $(a,b)$-integral in step 2.1, take uniform product grids with midpoint tags in the fixed parameter box. Each tagged sum has the separated form ${\sum }_C\mathrm{\mid }C\mathrm{\mid }\mathrm{\Phi }(a_C,b_C){\rho }_{{\epsilon }_j}(x-a_C){\sigma }_{{\epsilon }_j}(y-b_C)$. Terms with zero coefficient are omitted. Every remaining tag is in the nonzero set of $\mathrm{\Phi }$, so its factor supports lie in the two fixed compact neighborhoods defining $L$, even if the parameter box itself is not contained in $U\times V$. For each $j\ge 1$, choose the least grid level for which the error from $I_{{\epsilon }_j}$ in all target derivatives of total order at most $j$ is less than $1\mathrm{/}j$. Such a level exists by the uniform derivative convergence of those tagged sums established in step 2.1. The least-level rule is a defined integer, with no countable choice. Call the resulting finite product sum $P_j$.

For fixed $\gamma$, the difference $D^{\gamma }(P_j-\mathrm{\Phi })$ is bounded, for $j\ge \max (1,\mathrm{\mid }\gamma \mathrm{\mid })$, by $1\mathrm{/}j+\sup \mathrm{\mid }{D}^{\gamma }(I_{{\epsilon }_j}-\mathrm{\Phi })\mathrm{\mid }$, which tends to zero by step 3.1. All supports are in $L$, so F6 gives $P_j\to \mathrm{\Phi }$ in $\mathcal{D}(U\times V)$. If either domain is empty only the zero test occurs, covered in step 1.1. All integrations were of continuous compactly supported Riemann integrands and all grids were specified, so no choice axiom entered. $\mathrm{\square }$