Letting $\varphi(t)$ be the characteristic function for the probability measure $\mu$, we know if $\int \left|\varphi(t)\right|dt < \infty$, then $\mu$ has density function
$$f(y) = \frac{1}{2\pi} \int e^{-ity} \varphi(t)dt.$$
Example 3.3.8 states that what is denoted as "Polya's distribution", its pdf is $\frac{1-\cos x}{\pi x^2}$ and its characteristic function is $(1-\left|t\right|)^+.$
Now, there is a proof below. But I'm not certain what this is proving.
$\frac{2(1-\cos t)}{t^2}$ is the characteristic function for the triangular distribution(whose pdf is given by $1-\left|x\right|$ for $x \in (-1,1)$), so by theorem 3.3.5, it seems that
$$\frac{1}{2\pi} \int \frac{2(1-\cos s)}{s^2}e^{-isy} ds = (1-\left|y\right|), $$ not $(1-\left|y\right|)^+$.
I'm not even sure what the proof is proving.
I'd like to clear my misunderstanding.