Spectral alias identity for fixed regular-grid sampling

First establish the countability needed for the sum. By [F7], the quotient map $\pi :\mathbb{N}\times \mathbb{N}\to \mathbb{Z}$ is surjective, and $\beta :\mathbb{N}\to \mathbb{N}\times \mathbb{N}$ is bijective. Their composite is onto: for any class $z$, take a representative $(a,b)$ and the unique $j$ with $\beta (j)=(a,b)$; then $\pi (\beta (j))=z$. Thus $\mathbb{Z}$ is at most countable by the surjection criterion in [F7]. Its finite power ${\mathbb{Z}}^d$ is at most countable by [F7] and is nonempty, since it contains the constant tuple with value $[(0,0)]$ (and for $d=0$ the empty tuple). The same criterion supplies a surjection $e:\mathbb{N}\to {\mathbb{Z}}^d$. Put $K_m:=\{e(j):0\le j\le m\}$. These are increasing finite subsets whose union is ${\mathbb{Z}}^d$. For a nonnegative family $(a_k)$ define its sum as ${\lim }_m{\sum }_{k\in K_m}{a}_k$. Each index is counted once, even if $e$ repeats values. Every finite subset of ${\mathbb{Z}}^d$ is contained in some $K_m$: choose a preimage index for each of its finitely many elements and take their maximum. Hence the limit equals the supremum of all finite-subset sums and is invariant under reindexing. No countable family of choices is required. Each $\theta \mapsto f((\theta +2\pi k)\mathrm{/}\delta )$ is nonnegative Borel. The finite partial sums over $K_m$ are Borel and increase; for each real $a$, the strict superlevel set of their limit is the union of their Borel strict superlevel sets, so the limit is measurable. Reindexing the sum over finite subsets shows $g_{\delta }$ is periodic. The translates $Q+2\pi k$, $k\in {\mathbb{Z}}^d$, form a disjoint partition of ${\mathbb{R}}^d$. Put $P_m:={\bigcup }_{k\in K_m}(Q+2\pi k)$, so $P_m\uparrow {\mathbb{R}}^d$. Translation invariance [F2] first for indicators, then for simple functions by [F9] and by monotone convergence [F3] for nonnegative Borel functions, gives the cellwise integral translations. Monotone convergence [F3] then gives ${\int }_Q{g}_{\delta }(\theta )d\theta ={\delta }^{-d}{\lim }_m{\sum }_{k\in K_m}{\int }_{Q+2\pi k}f(u\mathrm{/}\delta )du={\delta }^{-d}{\int }_{{\mathbb{R}}^d}f(u\mathrm{/}\delta )du\mathrm{.}$ For the dilation identity, [F2] gives it first for indicators of Borel sets. For a nonnegative simple function written over disjoint level sets, summing these indicator identities gives the identity for the simple function by [F9]. By [F8], every nonnegative Borel function is an increasing limit of simple functions; monotone convergence [F3] therefore extends the dilation identity to every such function, including $f$. Thus ${\int }_Q{g}_{\delta }(\theta )d\theta ={\int }_{{\mathbb{R}}^d}f(\omega )d\omega <\mathrm{\infty }\mathrm{.}$ The measurable set where $g_{\delta }=\mathrm{\infty }$ is null: if it had positive measure, the integral would be at least $M$ times that measure for every $M>0$, contradicting finiteness. This set is periodic, so a finite-valued representative obtained by replacing $g_{\delta }$ by zero there remains periodic and defines the same measure.

For integer $h$, the identity $e^{ih\cdot (\theta +2\pi k)}=e^{ih\cdot \theta }$ and the finite mass in step 1.1 permit the same increasing finite-union partition for the absolutely integrable complex integrand. Translation on each cell, using the integral- translation identity established in step 1.1 on the nonnegative parts of its real and imaginary parts, gives ${\int }_Q{e}^{ih\cdot \theta }{g}_{\delta }(\theta )d\theta ={\delta }^{-d}{\int }_{{\mathbb{R}}^d}{e}^{ih\cdot u}f(u\mathrm{/}\delta )du\mathrm{.}$ To apply dilation to this complex integral, write $e^{ih\cdot u}f(u\mathrm{/}\delta )$ as the differences of its nonnegative cosine and sine parts. Apply the nonnegative scaling identity established in step 1.1 to $f(\omega )\max (\cos (\delta h\cdot \omega ),0)$, $f(\omega )\max (-\cos (\delta h\cdot \omega ),0)$, and the corresponding two sine parts. Each is Borel and bounded by $f$, hence integrable; recombining their integrals yields ${\int }_Q{e}^{ih\cdot \theta }{g}_{\delta }(\theta )d\theta ={\int }_{{\mathbb{R}}^d}{e}^{i\delta h\cdot \omega }f(\omega )d\omega =C(\delta h)=c(h)\mathrm{.}$ The density defines a finite Borel measure on the torus, Radon by [F5] and [F4]. By [F1] the sample is a stationary lattice field; symmetry of real covariance identifies $\mathrm{Cov}(Y_{j+h},Y_j)$ with $C(\delta h)$. Uniqueness in [F6] identifies that measure with its torus spectrum. The mass identity at $h=0$ is $C(0)$.

Apply [F6] to the sampled spectrum in step 2.1 to obtain the first periodogram expectation identity. Since ${\mathrm{\Phi }}_n$ is periodic, bounded for each fixed $n$, and nonnegative, partitioning the density integral exactly as in step 1.1 gives its second expression over ${\mathbb{R}}^d$. That expression integrates a periodic kernel against the original density and does not remove the noncentral translates.