The closed selected domain has smooth endpoint extensions

Since $v$ is temporal, $dv$ is nonzero everywhere. Thus $0$ is a regular value and [F5] supplies the smooth slice $\mathrm{\Sigma }$ when nonempty; its tangent space is $\ker dv$, as seen in the regular-level coordinates. Fix $c\in A$. The normalized flow from $(c,L_c\mathrm{/}dv(L_c))$ exists on a neighborhood of the finite closed interval ending at temporal time zero. Openness of the flow domain and smooth dependence give the same interval, with slight endpoint margins, for all initial points in a neighborhood of $c$ in $C$. Evaluating at the smooth varying elapsed time $-v(c^{\mathrm{\prime }})$ gives a smooth map $H(c^{\mathrm{\prime }})$ into $\mathrm{\Sigma }$: the flow identity is $v(H(c^{\mathrm{\prime }}))=v(c^{\mathrm{\prime }})-v(c^{\mathrm{\prime }})=0$. In the slice charts this is also a smooth map into $\mathrm{\Sigma }$, since the fixed zero coordinate can be dropped. Its restriction to $A$ is $F$ by uniqueness. Transversality is explicit in the normalized family, whose derivative of $v$ with respect to elapsed time is $1$; hence the regularity supplier [F3] applies as well.

The closed subset $A$ of compact $C$ is compact and Borel. Its local extensions cover it finitely by compactness. Applying [F3] to the smooth normalized-flow family, with the unique transverse zero-level hits from step 1.1, gives a countable cover by restrictions to smaller source coordinate balls with Lipschitz coordinate representatives in target slice charts. Each extension equals $F$ on its intersection with $A$, so these local continuous maps make $F$ continuous on $A$, hence Borel. If $A$ is empty, the empty extension cover gives all these conclusions. This step uses no Riemannian metric on $C$ and no restriction on the target dimension.

For the additional area conclusion, the supplied dimension $\dim E=4$ and [F5] make the nonempty slice $\mathrm{\Sigma }$ three-dimensional, and the supplied fixed metrics make $(C,h)$ a Riemannian surface and $(\mathrm{\Sigma },k)$ a Riemannian three-region. Thus [F4] applies to the Borel domain and countable local smooth extensions established above, defining the measurable a.e. Gram two-Jacobian and counting multiplicity relative to these same fixed metrics. If $\mathrm{\Sigma }$ is empty, then $D$ and $A$ are empty and the Jacobian and multiplicity conventions are vacuous. Endpoint focal degeneracy may cause rank loss but does not destroy the smooth extension or justify multiplicity one.