i have exams on this course coming up in two days (i should probably write this post earlier). i was hoping to read more and try to find a solution myself but unfortunately failed.

let f be a positive function, integrable over R. prove that
$\displaystyle

\sum_{n \in Z} \int_0^1 f(x+n) d \mu = \int_{-\infty}^{\infty} f(x) d \mu

$

and deduce that $\displaystyle \sum_{n \in Z} f(x+n) $ converges

prove that

this is based on the assumption that for all x in 0..pi/2 ( i have done this part)

also one mroe if p >-1 and q belongs to the natural number. prove that

(i have done this part)

deduce that
any thoughts would be appreciated! thank you!