Hi and thank you for your help.
I got stuck for the last days with this chapter in this book
https://ccrma.stanford.edu/~jos/mdft/Sampling_Theorem.html.
What I do not understand is
http://www.dsprelated.com/josimages_new/mdft/491x100ximg1812.png.pagespeed.ic.0qeZZN3ORw.png
Last equality says that by definition IDTFT is an integral over (-π/T, π/T), however IDTFT is by definition an integral over (-π, π). This fact appears actually in the equation immediately following in the same document (sorry I cannot post more than 2 links!).
The result is that x(t) is both an integral over (-π/T, π/T) (i.e. (-Ω /2, Ω /2)) and over (-π, π), integral of the same function.
Thank you for any help.