Thermal frequency cutoffs may be removed under the envelope

Put $q_m=\mathrm{\mid }{\chi }_m{\mathrm{\mid }}^2$, $h_m=(1-q_m)W$, $g_{m,t}=q_m{f}_t$, and $r_{m,t}=(q_m-1)f_t$. The real and imaginary parts of ${\chi }_m$ and $f_t$ are Borel by [F1], [F5] and [F7]. For ${\chi }_m=a_m+i{b}_m$, the polynomial $a_m^2+b_m^2$ gives Borel $q_m$: the coordinate tuple is measurable, the polynomial is continuous by [F8], and composition is measurable by [F7]. The same argument applied to the tuples of real coordinates of $q_m,f_t,W$ gives Borel $h_m$ and both real coordinates of $g_{m,t},r_{m,t}$; the coordinate criterion then gives their complex measurability. Modulus is continuous because [F11] gives $\mid \mathrm{\mid }z\mathrm{\mid }-\mathrm{\mid }w\mathrm{\mid }\mid \le \mathrm{\mid }z-w\mathrm{\mid }$, so their moduli are also Borel by [F7]. All these functions are finite-valued.

Since $\beta \omega >0$, [F10] gives $e^{\beta \omega }>e^0=1$ and hence $N_{\beta }(\omega )>0$. The formula in [F1] and the unit modulus of both phases give $\mathrm{\mid }{f}_t\mathrm{\mid }\le \mathrm{\mid }\kappa {\mathrm{\mid }}^2{\omega }^2[N_{\beta }+(N_{\beta }+1)]=W$ by [F11]. Since $0\le q_m\le 1$, we obtain $\mathrm{\mid }{g}_{m,t}\mathrm{\mid }\le W$ and $\mathrm{\mid }{r}_{m,t}\mathrm{\mid }\le h_m\le W$. Step 1.1 supplies measurability, so monotonicity in [F9] gives $\int \mathrm{\mid }{g}_{m,t}\mathrm{\mid }\le M$, $\int \mathrm{\mid }{r}_{m,t}\mathrm{\mid }\le M$, and $\int h_m\le M$. Thus these cutoff products and error densities are integrable by [F9]; $f_t$ is integrable by [F1].

By linearity on the integrable functions just established and by the integral triangle inequality, $\mid \int q_m{f}_t-\int f_t\mid =\mid \int r_{m,t}\mid \le \int \mathrm{\mid }{r}_{m,t}\mathrm{\mid }\le \int h_m$. The last comparison uses [F9]. This bound holds for every real $t$ with the same right-hand side.

The hypothesis $q_m\to 1$ almost everywhere and finiteness of $W$ give $h_m\to 0$ almost everywhere. The functions $h_m$ and the zero limit are measurable, and $0\le h_m\le W$ with $\int W=M<\mathrm{\infty }$. Hence [F2] yields $\int h_m\to 0$. Together with step 3.1, this is uniform convergence on the entire real time axis with the asserted error bound.

For every positive integer $m$, the interval $[1\mathrm{/}m,m]$ is closed in $I$: its complement is the union of two relatively open intervals. It is therefore Borel by [F5]. Its real indicator is measurable by [F6], and adjoining its identically zero imaginary coordinate makes it Borel into $\mathbb{C}\simeq {\mathbb{R}}^2$ by [F7]. It is bounded by one, and for each $\omega >0$ every sufficiently large integer $m$ satisfies $1\mathrm{/}m\le \omega \le m$; thus its squared modulus converges pointwise to one. These shell cutoffs satisfy all hypotheses of the preceding steps. If $W$ is nonintegrable, the domination hypothesis used here fails and the lemma makes no claim about unregulated or distributional limits.