Fourier transform of the Bessel Function $J_0(x)$

Question

In his answer to Bitrex's question, Sasha used generalized function properties to derive the Fourier transform of order $0$ Bessel function of the first kind.

I can't work out the logic he used in his answer.
Can someone please clarify this for me?
Maybe mention the identities used in the derivation of the answer?
I'm Mainly puzzled by the transition to the third row, and the notation in the following, fourth row: −1≤ω≤1 .

Answer 1

In a distributional sense the following identity holds

$$\frac1{2\pi}\int_{\Bbb R}e^{ikx}\mathrm dx=\delta(k)$$

This is a standard result (I found this PDF after a quick search proving the identitiy and explaining the necessitiy of this result). In the linked answer $k=\omega+\sin\tau$.

${\bf 1}_{-1\leq\omega\leq1}$ is a so-called characteristic function. This one simply translates to

$${\bf 1}_{-1\leq\omega\leq1}=\begin{cases}1&,\text{if }-1\leq\omega\leq1\\ 0&,{\text{else}}\end{cases}$$

I am not completey sure but it seems to me like ${\bf 1}_{-1\leq\omega\leq1}$ is introduced for reasons of convergence in the linked answer.