= Solution
Let $\mu_n$ be the <uniform spanning tree> measure on the finite connected induced graph $G_n$. Uniform spanning-tree edge indicators are negatively associated: increasing events depending on disjoint edge sets have nonpositive covariance. Together with the spatial Markov property, this implies the free-boundary monotonicity under graph enlargement: for every increasing cylinder event $A$, $\mu_n(A)$ is eventually nonincreasing once $G_n$ contains all edges on which $A$ depends.
Define the <free uniform spanning forest> measure by
$$
\mu_F(A)=\lim_{n\to\infty}\mu_n(A)
$$
for increasing cylinder events. These limits determine a unique probability measure, independent of the exhaustion; equivalently, $\mu_n$ converges weakly to $\mu_F$ on the product space.
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