An absolute thermal envelope controls radial correlations

The interval $I$ is open in $\mathbb{R}$, so [F9] makes each $A\in \mathcal{B}(I)$ an ambient Borel set: $A=I\cap B$ for some $B\in \mathcal{B}(\mathbb{R})$. Thus [F8] defines $\mu (A)=\lambda (A)$. The empty set has measure zero, and for any pairwise disjoint sequence in $\mathcal{B}(I)$ its union is still in $\mathcal{B}(I)$ and countable additivity of $\lambda$ gives countable additivity of $\mu$. Hence the stated restriction is a measure on the interval, as needed by the integral suppliers.

First, [F19] gives continuity of composition: if $h$ and $g$ are continuous and $V$ is open in the target, then $(g\circ h{)}^{-1}(V)=h^{-1}(g^{-1}(V))$ is open. The coordinate projections on ${\mathbb{R}}^n$ are continuous by [F20], and their restrictions to subspaces are continuous by [F14]. Identity and constant maps are continuous by [F19]. Therefore [F12] makes $\omega \mapsto \beta \omega$ continuous, and [F20] makes $\omega \mapsto (0,\pm \omega t)\simeq \pm i\omega t$ continuous into $\mathbb{C}\simeq {\mathbb{R}}^2$. Composition with [F11], followed by [F12], shows that $N_{\beta }$ and $\omega \mapsto e^{\pm i\omega t}$ are continuous on $I$; the denominator is nonzero because $e^{\beta \omega }>e^0=1$. Set $u=\mathrm{Re}\kappa$ and $v=\mathrm{Im}\kappa$, so $\mathrm{\mid }\kappa {\mathrm{\mid }}^2=u^2+v^2$ by [F17]. By [F10], $u,v$ are Borel and $\omega \mapsto (\omega ,u(\omega ),v(\omega ))$ is Borel into ${\mathbb{R}}^3$. On $I\times {\mathbb{R}}^2$ with its usual subspace topology, the expression for $W$ with $\mathrm{\mid }\kappa {\mathrm{\mid }}^2$ replaced by the sum of squares of the last two coordinates is continuous by the restricted coordinate projections, composition and [F12]. The real and imaginary parts of each phase are continuous coordinate projections of its continuous complex-valued map; the real and imaginary parts of the corresponding expression for $f_t$ are finite sums and products of these real-valued maps and the preceding continuous factors, hence are continuous by [F12]. Since the Borel map takes values in $I\times {\mathbb{R}}^2$, the trace identity [F9] also makes it Borel with that subspace codomain. Composing with the three continuous real-valued expressions gives Borel maps by [F10]. Its coordinate criterion then gives measurability of $f_t$ into $\mathbb{C}\simeq {\mathbb{R}}^2$; $W$ is measurable and nonnegative.

Since $N_{\beta }\ge 0$ and both phases have modulus one, multiplicativity and the triangle inequality in [F18] give $\mathrm{\mid }{f}_t\mathrm{\mid }\le \mathrm{\mid }\kappa {\mathrm{\mid }}^2{\omega }^2[N_{\beta }+(N_{\beta }+1)]=W$. By [F4], $\int \mathrm{\mid }{f}_t\mathrm{\mid }\le M<\mathrm{\infty }$, so [F1] gives absolute integrability, and [F2] gives $\mathrm{\mid }\int f_t\mathrm{\mid }\le M$. At $t=0$ both phases equal one, so $f_0=W$ pointwise and $\int f_0=M$. This pointwise identity also shows, independently of the finiteness assumption, that the nonnegative equal-time integral is finite exactly when $\int W$ is finite.

Let $t_n\to t$ be any real sequence. For each fixed $\omega$, the continuous coordinate operations and composition used in step 1.2, now with variable $t$, and the exponential in [F11] give $f_{t_n}(\omega )\to f_t(\omega )$ for every $\omega \in I$. Steps 1.2 and 2.1 supply measurable functions and the single integrable majorant $W$, so [F5] gives $\int f_{t_n}\to \int f_t$ (equivalently, the integrals of $f_{t_n}-f_t$ tend to zero by [F3]). The domain $\mathbb{R}$ and codomain $\mathbb{C}$ carry their usual metrics, and Countable Choice is assumed. Therefore [F13] converts this sequential convergence for every such sequence into continuity of $t\mapsto \int f_t$, completing the claims.