Polya's Condition for characteristic functions

Im reading the proof of the Pólya's condition ( Theorem 4.3.1) in the book "Characteristic Functions , by Eugene Lukacs 2ed".

But i dont understand the following statament in the proof:

Asume that: (Because i'm understand at this point)

$$p(x)=\frac{1}{\pi x}\int_0^\infty g(t)\sin(tx) dt $$

with $g(t)$ is non-increasing , non negative function, for $t> 0$ and $$\lim_{t\to\infty} g(t) = 0 $$

Then $$p(x) = \frac{1}{\pi x} \int_{0}^{\pi/x} \[ \sum_{j=0}^{\infty}(-1)^j g(t+\frac{j\pi}{x}) \] \sin(tx) $$

I dont see the final equality , the serie is a approximation or just a smart rewriter of g???