Given $Z$ is an integer-valued r.v., Its mass function is:
$$P(Z=z) = \int_{-\pi}^{\pi}\frac{\exp(-itz)}{2\pi}\times\phi_Z(t)\;dt, \quad z\in\mathbb{Z}$$
I am clear about how to show this by using inversion, but still have no idea about the method without general inversion theorem. Anyone may give some aid?