Measurable domains on smooth initial surfaces

Work in ZFC. Let $(C,h)$ be a smooth Hausdorff second-countable Riemannian surface. A measurable surface domain is a Borel subset $A\in \mathcal{B}(C)$, where $\mathcal{B}(C)$ is generated by the manifold-open sets; no openness, smooth boundary or Lipschitz boundary is required. Let $v_h$ be the intrinsic Borel measure of the Riemannian density $\sqrt{\det (h_{ab})}\mathrm{\mid }d{x}^{1}d{x}^2\mathrm{\mid }$. Its completion is denoted ${\mu }_C$: its domain consists of $B\cup N$ with $B,Z\in \mathcal{B}(C)$, $v_h(Z)=0$ and $N\subseteq Z$, and ${\mu }_C(B\cup N)=v_h(B)$. For a measurable surface domain write $\mathrm{\mid }A\mathrm{\mid }={\mu }_C(A)$. A closed subset of $C$ is an admissible Borel domain; it need not itself be a surface.

For $(X,g)=(C,h)$ or a smooth Hausdorff second-countable Riemannian three-manifold $(\mathrm{\Sigma },k)$, work separately on each connected component $X_j$. Give $X_j$ the finite Riemannian distance $d_{g,j}(p,q)={\inf }_{\gamma }{L}_g(\gamma )$ over piecewise $C^1$ curves in $X_j$ joining $p$ to $q$. There are at most countably many components. Set $c_2={\mathcal{H}}_{\mathrm{u}\mathrm{n}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m},{\mathbb{R}}^2}^2((0,1{]}^2)>0$, using the unnormalised diameter-cover construction, and for every $E\subseteq X$ define ${\mathcal{H}}_g^2(E)=c_2^{-1}{\sum }_j{\mathcal{H}}_{\mathrm{u}\mathrm{n}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m},(X_j,d_{g,j})}^2(E\cap X_j)$. This is an outer value on arbitrary sets; as a measure we use the complete componentwise Carathéodory domain, which includes all Borel sets. Thus for $D\subseteq \mathrm{\Sigma }$ write $\mathrm{\mid }D\mathrm{\mid }={\mathcal{H}}_k^2(D)$. The normalization equals ordinary Euclidean area on planes.

On $C$, ${\mathcal{H}}_h^2(A)={\mu }_C(A)$ for Borel $A$ and for every set in the completed area domain. If $S\subseteq \mathrm{\Sigma }$ is a smooth embedded surface with induced tensor $h_S=k{\mathrm{\mid }}_{TS}$, then ${\mathcal{H}}_k^2(E)={\mu }_S(E)={\mathcal{H}}_{h_S}^2(E)$ for Borel $E\subseteq S$ and their area-completed sets. Here the first Hausdorff value uses the ambient component distance of $\mathrm{\Sigma }$, whereas the last uses the intrinsic component distances of $S$. No equality of these distances is asserted.

A subset $E\subseteq X$, with $X$ and $g$ as above, is countably two-rectifiable if there are a set $N\subseteq E$ with ${\mathcal{H}}_g^2(N)=0$ and countably many maps $f_m:B_m\subseteq {\mathbb{R}}^2\to X_{j(m)}$ such that $E\setminus N\subseteq {\bigcup }_m{f}_m(B_m)$. Each $f_m$ is Lipschitz from Euclidean distance on $B_m$ to the specified finite distance $d_{g,j(m)}$ on its target component. No measurable-domain or boundary-regularity requirement is imposed on the parameter subsets $B_m$.

Given: The objects and hypotheses in the Definition, with the suppliers recorded above.

1.1 Every component of a manifold is open: connected coordinate balls about each of its points remain in that component. A countable base assigns to each component its least-indexed contained basis member, so there are at most countably many components. Apply the connected-distance suppliers on each component, with precisely their connectedness hypothesis. A set is Borel on the whole manifold exactly when its restrictions to the components are Borel, since these components are a countable open partition with the distance topology. [F1, F4, given]

1.2 Fix $0<\epsilon <1$ and a point in a Riemannian surface. Alternatively fix a point of an embedded surface $S\subseteq \mathrm{\Sigma }$ and use an ambient slice chart. In either case choose linear coordinates centered at the point so that the metric matrix at the center is the identity; in the slice case choose the first two orthonormal coordinate vectors tangent to $S$, preserving its coordinate plane. Smoothness of the matrix entries gives a coordinate ball $B_R$ with closure in the chart on which $(1-\epsilon {)}^2\mathrm{\mid }v{\mathrm{\mid }}^2\le g_x(v,v)\le (1+\epsilon {)}^2\mathrm{\mid }v{\mathrm{\mid }}^2$. Explicitly, it suffices that every entry of $G_x-I$ has absolute value at most $(2\epsilon -{\epsilon }^2)\mathrm{/}n$, since $\mathrm{\mid }{v}^T(G_x-I)v\mathrm{\mid }\le n{\max }_{a,b}\mathrm{\mid }(G_x-I{)}_{ab}\mathrm{\mid }\mathrm{\mid }v{\mathrm{\mid }}^2$; continuity gives that bound on a sufficiently small closed ball. Here $n=2$ intrinsically and $n=3$ in the ambient slice calculation. [F1, F7, given, algebra]

2.1 The positive smooth Riemannian density glues independently of charts and gives $v_h$ with the displayed coordinate formula. The stated completion construction therefore defines ${\mu }_C$ on exactly its specified domain. Closed sets are Borel because their complements are open. Apply the Hausdorff suppliers componentwise: multiplication by $c_2^{-1}$ and countable summation preserve outer subadditivity, Borel additivity and completeness on the componentwise measurable domain. The Euclidean normalization theorem gives normalized value ${\lambda }_2^{\ast }$ on ${\mathbb{R}}^2$, and hence area on any Euclidean plane, since isometries preserve the diameter-cover costs in both directions. No curved-surface identity is inferred from normalization alone. [A1, F1, F2, F3, F5, step 1.1]

2.2 Choose $0<r<R\mathrm{/}3$. For points with coordinates $x,y\in B_r$, the straight segment stays in $B_R$ and has metric length at most $(1+\epsilon )\mathrm{\mid }x-y\mathrm{\mid }$. Every joining curve staying in $B_R$ has length at least $(1-\epsilon )\mathrm{\mid }x-y\mathrm{\mid }$. If a joining curve leaves $B_R$, its initial segment to the first exit has coordinate displacement at least $R-r>2r\ge \mathrm{\mid }x-y\mathrm{\mid }$, so its length has the same lower bound. This uses continuity for the first exit, the metric-speed bounds and the coordinate displacement integral estimate; it applies to every piecewise $C^1$ joining curve, without existence of a minimizer. Taking the infimum gives $(1-\epsilon )\mathrm{\mid }x-y\mathrm{\mid }\le d_g(x,y)\le (1+\epsilon )\mathrm{\mid }x-y\mathrm{\mid }$. In a slice chart restrict $x,y$ to the surface coordinate plane: its straight segment lies in $S$, while the lower estimate already holds for all ambient joining curves. Thus the restricted ambient distance has these same bounds, even when curves outside $S$ are shorter than intrinsic curves. [F4, F7, F8, step 1.2]

3.1 Let $U$ be the small surface patch in step 2.2 and let $B\subseteq U$ be Borel in the surface. Denote its two-dimensional coordinate image by $x(B)$. Applying Lipschitz control in both directions to the distance bounds gives $(1-\epsilon {)}^2{\lambda }_2(x(B))\le H(B)\le (1+\epsilon {)}^2{\lambda }_2(x(B))$, where $H$ denotes either intrinsic normalized Hausdorff area or the ambient normalized Hausdorff value in the slice case. The two eigenvalues of the induced surface matrix lie between $(1-\epsilon {)}^2$ and $(1+\epsilon {)}^2$, so its area density lies between those same bounds. The chart-restriction formula for $v_h$ consequently gives $a_{\epsilon }{v}_h(B)\le H(B)\le a_{\epsilon }^{-1}{v}_h(B)$, with $a_{\epsilon }=((1-\epsilon )\mathrm{/}(1+\epsilon ){)}^2$. All constants are positive, so these comparisons also apply to infinite values. [F2, F5, F6, F7, step 1.2, step 2.2, algebra]

4.1 For each fixed $\epsilon$, second countability supplies a countable cover by these small surface patches. Subtract earlier patches to obtain a disjoint Borel partition. In the embedded case each slice patch is relatively closed in an ambient open coordinate neighborhood, hence ambient Borel; a surface-Borel subset of such a patch is ambient Borel as well. Thus both the intrinsic and ambient Hausdorff restrictions are countably additive on the pieces. Summing step 3.1 proves the same two-sided comparison on every Borel surface set. Let $\epsilon \downarrow 0$ to get equality: for finite area the constants tend to one, and for infinite area the lower bound forces infinite Hausdorff value. Componentwise sums treat disconnected surfaces and ambient manifolds. This proves both Borel identities stated in the Definition. [A1, F1, F2, F5, F7, F9, step 1.1, step 3.1]

5.1 For a set in the area completion, write it as $B\cup N$ with $N$ contained in an area-null Borel set $Z$. Step 4.1 makes $Z$ null for each compared Hausdorff convention, whose complete measure therefore measures $N$ with value zero. Its value on $B\cup N$ is the area-completed value $v_h(B)$. The rectifiability definition uses precisely these outer-null sets and the finite component metrics; it asserts no parametrization theorem. All original area conventions, closed-domain admissibility and the absence of boundary regularity requirements are retained. [F3, F5, F6, step 2.1, step 4.1] ∎

The constant $c_2$ is defined by the unnormalised diameter-cover construction, not by a numerical identification of that constant. The Euclidean normalization is supplied by the Euclidean Hausdorff theorem; the curved and ambient surface agreement is the local argument above. No endpoint area formula or article claim is used to justify this prerequisite.