Kigami's construction of Laplacians on post-critically finite fractals

Statement

Let $K \subset \R^d$ be the attractor of an iterated function system of contracting similitudes $F_1, \dots, F_N$, and assume that $K$ is connected and post-critically finite. Let $V_0 \subset K$ be its finite boundary set, let $V_{m+1} = \bigcup_{i=1}^N F_i(V_m)$, and for a word $w = w_1 \cdots w_m$ write $F_w = F_{w_1} \circ \cdots \circ F_{w_m}$ and $r_w = r_{w_1} \cdots r_{w_m}$. Let $\mathcal{E}_0(u) = \sum_{\set{p, q} \subset V_0} c_{pq} (u(p) - u(q))^2$ be the energy of an electrical network on $V_0$ with conductances $c_{pq} = c_{qp} \ge 0$ that connect $V_0$, and let $r_1, \dots, r_N \in (0, 1)$. For $u \colon V_m \to \R$ set $$\mathcal{E}_m(u) = \sum_{\abs{w} = m} r_w^{-1} \, \mathcal{E}_0(u \circ F_w|_{V_0}).$$ Assume the renormalization condition: for every $u \colon V_0 \to \R$, the minimum of $\mathcal{E}_1(v)$ over all $v \colon V_1 \to \R$ with $v|_{V_0} = u$ equals $\mathcal{E}_0(u)$. Then:

1. For every function $u \colon K \to \R$, the sequence $\mathcal{E}_m(u|_{V_m})$ is nondecreasing in $m$; let $\mathcal{E}(u)$ be its limit in $[0, \infty]$.
2. Let $\mathcal{F} = \set{u \colon K \to \R : u \text{ continuous}, \ \mathcal{E}(u) < \infty}$, and define $\mathcal{E}(u, v)$ from $\mathcal{E}$ by polarization. Then $(\mathcal{E}, \mathcal{F})$ is a resistance form on $K$, and its resistance metric induces the usual topology of $K$.
3. For every finite Borel measure $\mu$ on $K$ whose support is $K$, $(\mathcal{E}, \mathcal{F})$ is a local regular Dirichlet form on $L^2(K, \mu)$.
4. The Laplacian $\Delta_\mu$ is defined by: $\Delta_\mu u = f$, with $u \in \mathcal{F}$ and $f$ continuous, if $\mathcal{E}(u, v) = -\int_K f v \, d\mu$ for every $v \in \mathcal{F}$ that vanishes on $V_0$.