= Solution
Use the usual implicit convention that the infinite connected <graph> is a <locally finite graph>. This is needed for the <simple random walk> and the finite wired constructions to be defined. Choose finite vertex sets $V_1\subset V_2\subset\cdots$ exhausting $V$, with each <induced subgraph> $G[V_k]$ connected.
For the <free uniform spanning forest>, sample a <uniform spanning tree> of $G[V_k]$, using only the internal edges. Its limiting law on edge subsets of $G$ is the <free uniform spanning forest> measure:
$$
\boxed{\mathrm{FSF}=\lim_{k\to\infty}\operatorname{UST}(G[V_k]).}
$$
The limit is <weak convergence of probability measures> on $\{0,1\}^E$, equivalently convergence of every finite-edge event.
For the <wired uniform spanning forest>, form $G_k^{\mathrm w}$ by identifying every vertex outside $V_k$ to one boundary vertex $\partial_k$. Retain all edges with at least one endpoint in $V_k$, preserve parallel edges, and discard loops at $\partial_k$. Sample a <uniform spanning tree> of this finite multigraph and restrict it to the original internal edges. The limiting law is
$$
\boxed{\mathrm{WSF}=\lim_{k\to\infty}\operatorname{UST}(G_k^{\mathrm w})\big|_E.}
$$
For each fixed finite set of edges the restriction is unambiguous once $k$ is large. The standard exhaustion theorem guarantees that both limits exist and do not depend on the chosen connected exhaustion. Both are random spanning <forests>; despite their names, they need not be connected.
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