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The capacity of a finite set is the total mass of its equilibrium measure of a finite set:Capacity is monotone under inclusion, though strict inclusion need not give strict inequality.

For a transient random walk on a weighted graph, let $A$ be a finite set of graph vertices, $H_A=\inf \{n\ge 0:X_n\in A\}$ and $T_A^{+}=\inf \{n\ge 1:X_n\in A\}$. Its last-exit decomposition for a transient random walk iswhere $g$ is the Green function of a transient weighted graph and $e_A$ is the equilibrium measure of a finite set.

On the event of a visit to $A$, transient graph and finiteness of $A$ ensure a unique last visit. Partition by its time $n$ and location $a$ and apply the Markov property at that deterministic time. Every summand is nonnegative, so Tonelli's theorem justifies the infinite sum. The identity expresses a hitting probability as occupation weighted by an escape probability; it does not require a last-visit time to be a stopping time. For a singleton $A=\{o\}$ it reduces to the singleton equilibrium potential from the Green function.

The displayed identity is false with the printed non-strict inequality. For example, take $A=B=\{y\}$ and let $r=P_y({\tilde{H}}_y<\infty )\in (0,1)$. The event $L_y\le {\tilde{H}}_y$ says that the walk returns to $y$ at most once, so its left side is ${\mu }_y(1-r^2)$, whereas its right side is $e_{\{y\}}(y)={\mu }_y(1-r)$.

The standard and evidently intended last-exit decomposition for a transient random walk has $L_B<{\tilde{H}}_A$. Decompose that corrected event according to $n=L_B$ and $z=X_n\in B$. The Strong Markov property at time $n$ givesReversibility of the random walk on a graph gives the path-reversal identitySince $e_B(z)={\mu }_z{P}_z({\tilde{H}}_B=\infty )$ is the equilibrium measure of a finite set, summing first over $n$ and then over $z$ yields