Finite lattice gauge data define an invariant probability

For $h,k\in G^{\mathcal{V}}$, applying $k$ after $h$ sends an edge $e$ to $k_{s(e)}{h}_{s(e)}{U}_e{h}_{t(e)}^{-1}{k}_{t(e)}^{-1}=(kh{)}_{s(e)}{U}_e(kh{)}_{t(e)}^{-1}$; the identity tuple acts identically. Hence the formulas define a group action. The finite powers have the product smooth structures specified in the Definition, so each coordinate is a composition of smooth projections, multiplication, and inversion maps; the action map is smooth. By [F3], its differential sends the specified tangent vector at the identity of $G^{\mathcal{V}}$ to a tangent vector at $U$; applying $d{f}_U$ therefore defines the stated directional derivative on every smooth $f$. The inherited axiom base is the stated ${\mathrm{A}\mathrm{C}}_{\omega }$ [A1].

Around a face word based at $v_0$, the intermediate factors $h_{v_i}^{-1}{h}_{v_i}$ cancel, leaving $U_p(h\cdot U)=h_{v_0}{U}_p(U)h_{v_0}^{-1}$. Thus each ${\mathrm{\Phi }}_p(U_p)$ and the total action are unchanged. If ${\mathcal{E}}^{+}=\mathrm{\varnothing }$, the link space is the singleton empty tuple with unit point mass, and the action preserves this measure. Otherwise, for fixed $h$, the inverse image of a measurable rectangle ${\prod }_e{A}_e$ under the link action is the rectangle ${\prod }_e(h_{s(e)}^{-1}{A}_e{h}_{t(e)})$; bi-invariance preserves each factor measure. The product-measure rectangle formula and uniqueness therefore show that the finite link product measure is gauge invariant on the full product sigma-algebra.

For each face $p$ and each $g\in G$, continuity of ${\mathrm{\Phi }}_p$ gives an open neighborhood on which $\mathrm{\mid }{\mathrm{\Phi }}_p\mathrm{\mid }\le \mathrm{\mid }{\mathrm{\Phi }}_p(g)\mathrm{\mid }+1$. Compactness of $G$ gives a finite subcover, so each ${\mathrm{\Phi }}_p$ is bounded; summing over the finite face set gives $\mathrm{\mid }{S}_{\mathrm{\Lambda }}(U)\mathrm{\mid }\le M$ for some finite $M$. By [F7], the action, $S_{\mathrm{\Lambda }}$, and its density are measurable; hence $e^{-M}\le e^{-S_{\mathrm{\Lambda }}(U)}\le e^M$. Since the product link measure is a probability, $e^{-M}\le Z_{\mathrm{\Lambda }}\le e^M$; in particular $0<Z_{\mathrm{\Lambda }}<\mathrm{\infty }$. The nonnegative density $Z_{\mathrm{\Lambda }}^{-1}{e}^{-S_{\mathrm{\Lambda }}}$ therefore defines the normalized probability measure ${\mathbb{P}}_{\mathrm{\Lambda }}$. Its invariance follows from the invariance of the product measure and of $S_{\mathrm{\Lambda }}$ in step 2.1.

For each $h\in G^{\mathcal{V}}$, define $T_h[f]:=[U\mapsto f(h^{-1}\cdot U)]$ on complex $L^2({\mathbb{P}}_{\mathrm{\Lambda }})$, with norm $\mathrm{\parallel }[f]{\mathrm{\parallel }}_2=(\int \mathrm{\mid }f{\mathrm{\mid }}^{2}d{\mathbb{P}}_{\mathrm{\Lambda }}{)}^{1\mathrm{/}2}$. Step 3.1 makes this well-defined and preserves the norm: invariance gives equality of integrals first for nonnegative simple functions and then, by the defining supremum for the nonnegative integral, for $\mathrm{\mid }f{\mathrm{\mid }}^2$. Thus each $T_h$ is a linear isometry. The invariant sector is exactly ${\bigcap }_{h\in G^{\mathcal{V}}}\ker (T_h-I)$; each $T_h-I$ is continuous by the norm triangle inequality, so its kernel is closed, and an arbitrary intersection of closed sets is closed.