In terms of analog signals, we can represent digital signal as :
$$ x[n] \triangleq x_{a}(nT) = \int_{-\infty}^{\infty}X_{a}(f) \, e^{j2\pi f nT} \ \mathrm{d}f $$
While if we focused on the integral on the right side and according to the Digital Signal Processing by John Proakis, chapter 6.1, we can rewrite it into :
$$ \int_{-\infty}^{\infty}X_{a}(f) \, e^{j2\pi f nT} \ \mathrm{d}f = \sum_{k=-\infty}^{\infty} \int_{(k-1/2)F_{s}}^{(k+1/2)F_{s}}X_{a}(f) \, e^{j2\pi nf/F_{s}} \ \mathrm{d}f $$
where $ F_{s} \triangleq \frac1T $. My question is how the second equation comes up ? what does the interval of $(k-1/2)F_{s}$ to $(k+1/2)F_{s}$ means ?
Furthermore, it is stated in the book that "observing the $X_{a}(f)$ in the interval of $(k-1/2)F_{s}$ to $(k+1/2)F_{s}$ is identical to $X_{a}(f-kF_{s})$ in the interval of $-F_{s}/2$ to $F_{s}/2$". Is there any explanation or derivation how both things are identical ?
Thank you so much, hope that my question is clear enough