Compact test classes localize to a Cauchy neighborhood

Every volume-time level $S_c$ of $U$ is also a Cauchy set of $M$. Let $\gamma$ be any inextendible causal curve in $M$. It meets the Cauchy hypersurface $\mathrm{\Sigma }\subset U$ from the Cauchy-morphism hypothesis. Because $U$ is causally convex, ${\gamma }^{-1}(U)$ is an interval: if two parameter values lie in $U$, the causal segment between them lies in $U$. The restriction of $\gamma$ to this interval is inextendible in $U$; a limit at a boundary point of the interval in $U$ would, by openness and continuity, extend the interval, while a limit at an end of the original curve would contradict its inextendibility in $M$. Hence this restriction meets the Cauchy level $S_c$ of $U$ exactly once by [F1]. This proves the claim for every $c$.

Choose any $a<b$. The two open sets $I^{+}(S_a)$ and $I^{-}(S_b)$ cover $M$. To see this, take a maximal future timelike geodesic through any point $p\in M$ using [F6]. By step 1.1 it meets both Cauchy levels, and it meets $S_a$ before $S_b$: between the two crossings the curve lies in $U$ by causal convexity, where $t_U$ strictly increases from $a$ to $b$ [F1,F7]. If $p$ is strictly after the $S_a$ crossing then $p\in I^{+}(S_a)$; otherwise it is strictly before the $S_b$ crossing and $p\in I^{-}(S_b)$. Openness follows from [F5]. Apply [F4] to obtain smooth ${\theta }_{+},{\theta }_{-}:M\to [0,1]$ with ${\theta }_{+}+{\theta }_{-}=1$, $\mathrm{supp}{\theta }_{+}\subset I^{+}(S_a)$ and $\mathrm{supp}{\theta }_{-}\subset I^{-}(S_b)$. Set $f_{\pm }:={\theta }_{\pm }f$ and $K_{\pm }:=\mathrm{supp}{f}_{\pm }$; these are compact, and $K_{\pm }$ lie in the stated open sets.

If $K_{+}$ is nonempty, choose $a^{\mathrm{\prime }}>a$ with $K_{+}\subset I_M^{+}(S_{a^{\mathrm{\prime }}})$, as follows. The opens $I_M^{+}(S_c)$ for $c>a$ cover $I_M^{+}(S_a)$: a timelike curve from $S_a$ to a point of this latter set has a short initial segment inside the open $U$, on which $t_U$ increases above $a$ and crosses some level $c>a$ before reaching the endpoint. For a finite subcover of the compact $K_{+}$ choose $a^{\mathrm{\prime }}$ between $a$ and the least selected $c$; ordering of the levels then gives the asserted containment. This never evaluates $t_U$ outside $U$. The sets $A:=J_M^{-}(S_a)$ and $B:=J_M^{+}(S_{a^{\mathrm{\prime }}})$ are closed and disjoint: any causal segment from the later level to the earlier one would stay in $U$ and contradict strict increase of $t_U$. A smooth partition for $M\setminus A$ and $M\setminus B$ gives ${\rho }_{+}:M\to [0,1]$ equal to zero near $A$ and one near $B$, with $\mathrm{supp}{\rho }_{+}\subset M\setminus A$ and $\mathrm{supp}(d{\rho }_{+})\subset M\setminus (A\cup B)$. Every point in that transition set is causally between $S_a$ and $S_{a^{\mathrm{\prime }}}$, because both are Cauchy sets of $M$; causal convexity puts this transition set inside $U$. The larger shadow $H_{+}:=J_M^{+}(S_a)\cap J_M^{-}(K_{+})$ is compact in $M$: cover $K_{+}$ by finitely many open chronological pasts $I_M^{-}(q_j)$, use transitivity to enclose $H_{+}$ in ${\bigcup }_j(J^{-}(q_j)\cap J^{+}(S_a))$, compact by [F9], and use closedness from [F2] and [F3]. The shadow need not lie in $U$. Put ${\psi }_{+}:={\rho }_{+}{E}_{-}{f}_{+}$. Its closed support lies in $H_{+}$, so ${\psi }_{+}\in {\mathcal{D}}_g(M)$. Since ${\rho }_{+}=1$ near $K_{+}$, the product rule gives $g_{+}:=f_{+}-P{\psi }_{+}=-[P,{\rho }_{+}]E_{-}{f}_{+}$, with $[P,\rho ]u:=P(\rho u)-\rho Pu$. Its support lies in the transition set inside $U$ and in the compact shadow $H_{+}$, so $g_{+}$ has compact support in $U$. By [F8], $E{g}_{+}=E{f}_{+}$. If $K_{+}=\mathrm{\varnothing }$, set $g_{+}=0$.

The time-reversed construction applies to $K_{-}\subset I_M^{-}(S_b)$. In detail, the opens $I_M^{-}(S_c)$ for $c<b$ cover $I_M^{-}(S_b)$ by taking a short terminal timelike segment inside $U$; a finite subcover selects $b^{\mathrm{\prime }}<b$ with $K_{-}\subset I_M^{-}(S_{b^{\mathrm{\prime }}})$. A partition for the complements of the closed disjoint sets $A^{\mathrm{\prime }}:=J_M^{-}(S_{b^{\mathrm{\prime }}})$ and $B^{\mathrm{\prime }}:=J_M^{+}(S_b)$ gives ${\rho }_{-}=1$ near $A^{\mathrm{\prime }}$, ${\rho }_{-}=0$ near $B^{\mathrm{\prime }}$, and $\mathrm{supp}{\rho }_{-}\subset M\setminus B^{\mathrm{\prime }}$. Its derivative support lies causally between the two levels and hence inside $U$. The shadow $H_{-}:=J_M^{+}(K_{-})\cap J_M^{-}(S_b)$ is compact by the reversed [F9], a finite cover of $K_{-}$ by chronological futures, and closedness, just as in step 3.1. Define ${\psi }_{-}:={\rho }_{-}{E}_{+}{f}_{-}\in {\mathcal{D}}_g(M)$ and $g_{-}:=f_{-}-P{\psi }_{-}=-[P,{\rho }_{-}]E_{+}{f}_{-}$. The support of $g_{-}$ is closed, contained in $H_{-}$ and in the transition set inside $U$, so it is compact in $U$; [F8] gives $E{g}_{-}=E{f}_{-}$. The zero-support case uses $g_{-}=0$.

Let $g_U$ be the restriction to $U$ of $g:=g_{+}+g_{-}$, whose two summands have compact support in $U$ by steps 3.1 and 4.1. Then ${\iota }_{\ast }{g}_U=g$, and $Eg=E{g}_{+}+E{g}_{-}=E{f}_{+}+E{f}_{-}=Ef$ by those same steps and $f=f_{+}+f_{-}$. By [F8], $\ker E=P{\mathcal{D}}_g(M)$, so $E(f-{\iota }_{\ast }{g}_U)=0$ implies $f-{\iota }_{\ast }{g}_U\in P{\mathcal{D}}_g(M)$. This is the asserted equality of test classes.