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We study the boundary theory of a connected weighted graph $G$ from theviewpoint of stochastic integration. For the Hilbert space \HE ofDirichlet-finite functions on $G$, we construct a Gel'fand triple $S \ci{\mathcal H}_{\mathcal E} \ci S'$. This yields a probability measure$\mathbb{P}$ on $S'$ and an isometric embedding of ${\mathcal H}_{\mathcal E}$into $L^2(S',\mathbb{P})$, and hence gives a concrete representation of theboundary as a certain class of "distributions" in $S'$. In a previous paper, weproved a discrete Gauss-Green identity for infinite networks which produces aboundary representation for harmonic functions of finite energy, given as acertain limit. In this paper, we use techniques from stochastic integration tomake the boundary $\operatorname{bd}G$ precise as a measure space, and obtain aboundary integral representation as an integral over $S'$.

Gel'fand triples and boundaries of infinite networks