Transport of a lattice spectral measure to torus coordinates

If $B_1,B_2,\dots$ are pairwise disjoint Borel subsets of ${\mathbb{T}}^d$, then their images under the bijection $\mathrm{\Psi }$ are pairwise disjoint Borel sets by [F1]. Hence $F\text{\hspace{0.17em}⁣}\left({\bigcup }_{m\ge 1}{B}_m\right)=\mu \text{\hspace{0.17em}⁣}({\bigcup }_{m\ge 1}\mathrm{\Psi }(B_m))={\sum }_{m\ge 1}\mu (\mathrm{\Psi }(B_m))={\sum }_{m\ge 1}F(B_m)\mathrm{.}$ Thus $F$ is a positive measure, and $F({\mathbb{T}}^d)=\mu (\widehat{{\mathbb{Z}}^d})<\mathrm{\infty }$.

For Borel $B\subseteq {\mathbb{T}}^d$ and $\epsilon >0$, outer regularity of $\mu$ gives an open $U\supseteq \mathrm{\Psi }(B)$ with $\mu (U)<\mu (\mathrm{\Psi }(B))+\epsilon$. Then ${\mathrm{\Psi }}^{-1}(U)$ is open, contains $B$, and $F({\mathrm{\Psi }}^{-1}(U))=\mu (U)$, proving outer regularity of $F$. If $U$ is open in ${\mathbb{T}}^d$, inner regularity of $\mu$ gives compact $C\subseteq \mathrm{\Psi }(U)$ with $\mu (C)>\mu (\mathrm{\Psi }(U))-\epsilon$. Then ${\mathrm{\Psi }}^{-1}(C)$ is compact in $U$ and $F({\mathrm{\Psi }}^{-1}(C))=\mu (C)$, proving inner regularity on open sets. The torus is compact Hausdorff by [F4], so these properties make $F$ Radon as in [F2].

For an indicator $g={\mathbf{1}}_B$ of a Borel set, $\int gdF=F(B)=\mu (\mathrm{\Psi }(B))=\int {\mathbf{1}}_B({\mathrm{\Psi }}^{-1}(\chi ))\mu (d\chi )\mathrm{.}$ Finite linearity [F3] extends this equality to Borel simple functions. For a nonnegative Borel function $g$, the simple-approximation theorem [F3] supplies Borel simple functions $s_m\uparrow g$. Since ${\mathrm{\Psi }}^{-1}$ is continuous, $s_m\circ {\mathrm{\Psi }}^{-1}\uparrow g\circ {\mathrm{\Psi }}^{-1}$ measurably on the dual. Applying monotone convergence [F3] to the simple-function identity on both spaces gives the asserted identity for $g$. Splitting the real and imaginary parts of a bounded complex Borel function into positive and negative parts then gives the stated identity by [F5].