Global locally finite structure of distributions

Take a partition $({\eta }_i)$ from F3, indexed by positive integers or a finite initial segment, and discard zero functions. Write $S_i=\mathrm{supp}{\eta }_i$. These nonempty compacts admit a locally finite family of relatively compact open neighborhoods $V_i$: set ${\epsilon }_i=\min (1\mathrm{/}i,1,\mathrm{dist}(S_i,{\mathbb{R}}^n\setminus \mathrm{\Omega })\mathrm{/}2)$, interpreting distance to the empty set as infinity, and take $V_i=\{x:\mathrm{dist}(x,S_i)<{\epsilon }_i\}$. Their closures are compact inside $\mathrm{\Omega }$. To see local finiteness, fix a ball $B(x,2r)$ compactly inside $\mathrm{\Omega }$. Its closure meets only finitely many $S_i$ by the given local finiteness and compactness. The remaining supports lie outside this ball; for all sufficiently large $i$, ${\epsilon }_i<r$, so their $V_i$ miss $B(x,r)$. Only finitely many exceptions remain.

Define $u_i={\eta }_{i}u$. By F6 its support is contained in $S_i$, so F5 gives it some global order $m_i$. F1 gives a finite representation $u_i={\sum }_{\alpha }{\mathrm{\partial }}^{\alpha }{u}_{g_{i,\alpha }}$ with continuous coefficient functions compactly supported in $V_i$. Use F4 to select one such finite representation for every index, including an order and its coefficient tuple; the sets of possible tuples are nonempty by F1 and F5. Set $g_{i,\alpha }=0$ outside its finite index set.

Put $g_{\alpha }={\sum }_i{g}_{i,\alpha }$. Step 1.1 makes these sums locally finite, hence continuous. A locally finite union of closed coefficient supports is closed: near any point only finitely many supports occur, and the complement of that finite union is open. Consequently $\mathrm{supp}{g}_{\alpha }$ is contained in the union of the corresponding coefficient supports. A compact $H$ meets only finitely many $V_i$, and each of those indices has only finitely many coefficients, so $H$ meets only finitely many $\mathrm{supp}{g}_{\alpha }$.

Fix $\alpha$ and cover $\mathrm{\Omega }$ by open balls whose compact closures lie in $\mathrm{\Omega }$. Each closure meets only finitely many $V_i$, so on each ball the regular functional of $g_{\alpha }$ is the finite sum of the corresponding regular distributions $u_{g_{i,\alpha }}$. These local distributions agree on overlaps because both finite expressions integrate the same locally finite pointwise sum. F2 therefore glues them to a distribution on $\mathrm{\Omega }$, and uniqueness identifies that distribution with the regular functional denoted $u_{g_{\alpha }}$. For a test $\phi$, only finitely many partition supports meet its compact support, so the partition identity from F3 and linearity give $u(\phi )={\sum }_{i}u({\eta }_i\phi )={\sum }_i{u}_i(\phi )$. Insert the representations from step 2.1. The same compact meets only finitely many $V_i$, and all distribution derivatives of regular coefficients supported elsewhere vanish on the test. Thus both sums may be interchanged as finite sums, and finite linearity of the regular integral gives $u(\phi )={\sum }_{\alpha }({\mathrm{\partial }}^{\alpha }{u}_{g_{\alpha }})(\phi )$.

If $u$ has global order at most $m$, every $u_i={\eta }_{i}u$ has order at most $m$: on any fixed compact, the product rule bounds $p_m({\eta }_i\phi )$ by a finite constant times $p_m(\phi )$, without increasing the order. Choose all representations in step 2.1 using this same $m$. Then F1 permits only the fixed finite index box ${\alpha }_i\le m+2$, proving the final assertion. Empty $\mathrm{\Omega }$ or $u=0$ permits all coefficients zero. The partition and open enlargements use no choice; the representation selection and its supplier have the explicit AC use. $\mathrm{\square }$