[proofplan] We use the Baouendi-Treves local propagation principle assumed in the statement as the local mechanism: on each admissible patch $P\in\mathscr P$, an $L^2_{\mathrm{loc}}$ distributional CR function that vanishes on a non-empty open piece must vanish on each smaller connected open subpatch in the same local CR orbit. We then propagate the zero set along the connected CR orbit by covering a piecewise CR path with finitely many admissible patches. Finally, an exhaustion by compact subsets converts local almost-everywhere vanishing into vanishing on all of $\mathcal O$. [/proofplan] [step:Define the almost-everywhere zero set and record its initial non-emptiness] Let $Z\subset\mathcal O$ denote the set of points $p\in\mathcal O$ for which there exists an open neighbourhood $V_p\subset\mathcal O$ of $p$ such that \begin{align*} f=0\quad \mathcal H^d\text{-a.e. on }V_p. \end{align*} By definition, $Z$ is open in $\mathcal O$. Since the hypothesis gives a non-empty [open set](/page/Open%20Set) $U\subset\mathcal O$ with $f=0$ $\mathcal H^d$-a.e. on $U$, every point of $U$ belongs to $Z$, and hence $Z\ne\varnothing$. [/step] [step:Use the Baouendi-Treves local propagation principle on admissible patches] We shall use the Baouendi-Treves local propagation principle assumed in the theorem statement; this local input is the propagation consequence associated with the [Baouendi-Treves approximation theorem](/theorems/9229) [citetheorem:9229]. [claim:Local uniqueness from Baouendi-Treves approximation] Let $P\in\mathscr P$, and let \begin{align*} g\in L^2_{\mathrm{loc}}(P,\mathcal H^d;\mathbb C) \end{align*} be a distributional CR function. If $g=0$ $\mathcal H^d$-a.e. on a non-empty open subset $A\subset P$, then $g=0$ $\mathcal H^d$-a.e. on every relatively compact connected open subpatch $P_0\Subset P$ contained in the local CR orbit of $A$ in $P$. [/claim] [proof] This is exactly the Baouendi-Treves local propagation principle assumed in the theorem statement, applied to the admissible connected generic CR coordinate patch $P\in\mathscr P$, the distributional CR function $g$, and the non-empty open zero set $A\subset P$. The conclusion of that principle gives $g=0$ $\mathcal H^d$-a.e. on every relatively compact connected open subpatch $P_0\Subset P$ contained in the local CR orbit of $A$ in $P$. [/proof] Apply this claim with $g=f|_P$ whenever $P\in\mathscr P$. The hypotheses are satisfied because $f$ is a distributional CR function on $\mathcal O$, restriction preserves the distributional CR equations, and $\mathcal H^d|_P$ is the induced smooth density class on the embedded patch. [guided] The point of this step is to isolate the only deep local input. Let $P\in\mathscr P$, and let \begin{align*} g\in L^2_{\mathrm{loc}}(P,\mathcal H^d;\mathbb C) \end{align*} be distributional CR. Suppose that $g=0$ $\mathcal H^d$-a.e. on a non-empty open subset $A\subset P$. The Baouendi-Treves local propagation principle in the theorem statement applies precisely to this data: $P$ is one of the admissible generic CR coordinate patches in $\mathscr P$, $g$ is an $L^2_{\mathrm{loc}}$ distributional CR function on $P$, and $A$ is a non-empty open subset of $P$ on which $g$ vanishes almost everywhere. Its conclusion gives \begin{align*} g=0\quad \mathcal H^d\text{-a.e. on }P_0 \end{align*} for every relatively compact connected open subpatch $P_0\Subset P$ contained in the local CR orbit of $A$ in $P$. Applying this to $g=f|_P$ is legitimate because restriction of a distributional CR function to an open CR submanifold patch is still distributional CR, and the measure used on $P$ is the restriction of $\mathcal H^d$. [/guided] [/step] [step:Show that the almost-everywhere zero set is closed under local CR propagation] Let $p\in Z$, and let $q\in\mathcal O$ be such that $p$ and $q$ lie in an admissible patch $P\in\mathscr P$ and in the same local CR orbit inside $P$. Choose an open neighbourhood $V_p\subset\mathcal O$ of $p$ such that \begin{align*} f=0\quad \mathcal H^d\text{-a.e. on }V_p. \end{align*} Define $A:=V_p\cap P$. Then $A$ is a non-empty open subset of $P$, contains $p$, and satisfies \begin{align*} f=0\quad \mathcal H^d\text{-a.e. on }A. \end{align*} Because $p\in A$ and $q$ lies in the same local CR orbit in $P$ as $p$, the point $q$ belongs to the local CR orbit of $A$ in $P$. By the additional local-orbit neighbourhood hypothesis in the theorem statement, there exists a relatively compact connected open subpatch $P_0\Subset P$ such that $q\in P_0$ and $P_0$ is contained in the local CR orbit of $A$ in $P$. By the local uniqueness claim applied to $g=f|_P$, one has \begin{align*} f=0\quad \mathcal H^d\text{-a.e. on }P_0. \end{align*} Taking $B:=P_0$, we obtain an open neighbourhood $B\subset P$ of $q$ on which $f$ vanishes $\mathcal H^d$-a.e. Hence $q\in Z$. Thus $Z$ propagates through every admissible local CR segment inside $\mathcal O$. [/step] [step:Propagate the vanishing along a finite chain inside the CR orbit] Fix $x\in\mathcal O$. Since $\mathcal O$ is a connected CR orbit, $x$ can be joined to some point $x_0\in Z$ by a piecewise $C^1$ curve \begin{align*} \gamma:[0,1]\to\mathcal O \end{align*} whose tangent vector lies in the real CR distribution whenever the derivative exists. This is precisely the reachability property defining a connected CR orbit. By the admissible patch compatibility assumed in the theorem statement, the curve $\gamma$ admits finitely many parameters \begin{align*} 0=t_0<t_1<\cdots<t_m=1 \end{align*} and patches $P_1,\dots,P_m\in\mathscr P$ such that $\gamma([t_{j-1},t_j])\subset P_j$, the points $\gamma(t_{j-1})$ and $\gamma(t_j)$ lie in the same local CR orbit of $P_j$, $x_0\in P_1$, and $x\in P_m$. Starting from $x_0\in Z$, the previous step propagates membership in $Z$ from $\gamma(t_{j-1})$ to $\gamma(t_j)$ for each $j$. Induction over $j=1,\dots,m$ gives $x=\gamma(1)\in Z$. Therefore $Z=\mathcal O$. [/step] [step:Conclude vanishing on compact subsets and on the whole orbit] Let $K\subset\mathcal O$ be compact. Since $Z=\mathcal O$, for every $x\in K$ there exists an open neighbourhood $V_x\subset\mathcal O$ such that \begin{align*} f=0\quad \mathcal H^d\text{-a.e. on }V_x. \end{align*} The compactness of $K$ gives points $x_1,\dots,x_r\in K$ such that \begin{align*} K\subset \bigcup_{j=1}^r V_{x_j}. \end{align*} For each $1\le j\le r$, define the null set \begin{align*} E_j:=\{y\in V_{x_j}: f(y)\ne 0\} \end{align*} after choosing a measurable representative of $f$ on $V_{x_j}$. Since $f=0$ $\mathcal H^d$-a.e. on $V_{x_j}$, one has \begin{align*} \mathcal H^d(E_j)=0. \end{align*} The exceptional set on $K$ is contained in the finite union $\bigcup_{j=1}^r E_j$, so finite subadditivity of $\mathcal H^d$ gives \begin{align*} \mathcal H^d\left(K\cap\{y\in\mathcal O:f(y)\ne 0\}\right)=0. \end{align*} Thus $f=0$ $\mathcal H^d$-a.e. on $K$. Conversely, if $f=0$ $\mathcal H^d$-a.e. on every compact set $K\subset\mathcal O$, choose a compact exhaustion $(K_j)_{j\in\mathbb N}$ of $\mathcal O$ by compact subsets with \begin{align*} \mathcal O=\bigcup_{j\in\mathbb N}K_j. \end{align*} [Countable subadditivity](/theorems/1108) then gives $f=0$ $\mathcal H^d$-a.e. on $\mathcal O$. The reverse implication is immediate by restriction from $\mathcal O$ to $K$. Hence the compact-set formulation and the global almost-everywhere formulation are equivalent, and the theorem follows. [/step]