= Solution
In the <voter model> on $\mathbb Z^2$, each site has a rate-one <Poisson process> clock; when it rings, that site copies one of its four neighbors, chosen uniformly. States are zero or one. For a <cylinder function> $f$, the local <infinitesimal generator> is
$$
Lf(\eta)=\sum_x\frac14\sum_{y\sim x}\bigl[f(\eta^{x\leftarrow y})-f(\eta)\bigr],
$$
where $\eta^{x\leftarrow y}$ agrees with $\eta$ except that the <voter model> opinion at $x$ is replaced by the <voter model> opinion at $y$. Only sites in the support of $f$ contribute, so this <infinitesimal generator> sum is finite.
The <voter-model duality> comes from independent <Poisson processes> of copying arrows of rate $1/4$ on each ordered neighboring pair. Trace the ancestors of specified sites backwards from time $t$: each ancestral line is a rate-one <continuous-time random walk> with uniform nearest-neighbor steps, and lines coalesce when they meet. The <voter model> opinions at time $t$ are the initial <voter model> opinions at those ancestral locations. This construction also explains the general duality theorem being used.
Two ancestral walks on $\mathbb Z^2$ meet almost surely. Until their meeting, their difference is a rate-two symmetric nearest-neighbor walk, and <simple random walk in two dimensions is recurrent>. If $\tau_{xy}$ is their meeting time, this gives
$$
P_\nu(\xi_t(x)\ne\xi_t(y))\le P(\tau_{xy}>t)\longrightarrow0
$$
for every initial law $\nu$. If $\nu$ is invariant, the left side is the fixed <probability> $\nu(\eta(x)\ne\eta(y))$, so it is zero. This holds for every pair. Countability then implies that $\nu$ is concentrated on configurations with all <voter model> opinions equal. Both constant configurations are <absorbing state>. Thus the <invariant measures of the two-dimensional voter model> are exactly
$$
\boxed{\nu=\alpha\delta_0+(1-\alpha)\delta_1,\qquad0\le\alpha\le1.}
$$
No translation-invariance assumption on $\nu$ was used.
For the last assertion, let the initial set of one-valued opinions $F$ be finite. Single-line duality gives
$$
P(\xi_t(x)=1)=P_x(X_t\in F)=\sum_{z\in F}p_t(x,z).
$$
Each <transition probability> tends to zero. One direct verification is the <Fourier transform> for the rate-one walk:
$$
p_t(x,z)=\frac1{(2\pi)^2}\int_{[-\pi,\pi]^2}
e^{-ik\cdot(z-x)}\exp\left[-t\left(1-\tfrac12(\cos k_1+\cos k_2)\right)\right]dk.
$$
The integrand has modulus at most one and tends to zero except at the single point $k=0$; <dominated convergence> proves the claim. For any finite observation set $K$, the <union bound> now gives
$$
P(\xi_t\text{ has a one somewhere in }K)
\le\sum_{x\in K}\sum_{z\in F}p_t(x,z)\longrightarrow0.
$$
Therefore every finite <voter model> opinion pattern converges in law to the all-zero pattern. These cylinder distributions determine <weak convergence of probability measures> in the compact <product topology>, proving \b[finite-seed local extinction in the <voter model>] and
$$
\boxed{\mathcal L(\xi_t)\Longrightarrow\delta_0.}
$$
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