Interchanging limits when computing the Fourier transform

Question

In the top answer of this post here why can we interchange the two limits $\lim_{R\rightarrow \infty}$ and $\lim_{\epsilon \rightarrow 0^+}$?

Answer 1

Note that due to the asymmetry of the integrand, the expression is equal to \begin{align}\lim_{R\to \infty} &\lim_{\epsilon \to 0^+}\int_\mathbb{R} 1_{\{|k| >\epsilon\}}\frac{\phi(k)}{ik}(e^{ikR}+e^{-ikR})\,dk\\ &=\lim_{R\to \infty} \lim_{\epsilon \to 0^+}\int_\mathbb{R} 1_{\{|k| >\epsilon\}}\frac{\phi(k)-1_{\{|k|\leq1\}}\phi(0)}{ik}(e^{ikR}+e^{-ikR})\,dk \\ &=\lim_{R\to \infty} \int_\mathbb{R}\frac{\phi(k)-1_{\{|k|\leq1\}}\phi(0)}{ik}(e^{ikR}+e^{-ikR})\,dk=0,\end{align} where in the end we use the Riemman-Lebesgue Lemma for the function $k\mapsto \frac{\phi(k)-1_{\{|k|\leq1\}}\phi(0)}{ik}$ , which is integrable.