Suppose that $X, Y, Z$ are independent random variables such that $X+Y$ and $X+Z$ have the same distribution. Is it necessary that $Y$ and $Z$ have the same distribution?
I tried to show that the characteristic functions of $Y$ and $Z$ are the same. But only what I could prove is that $E[e^{itY}]=E[e^{itZ}]$ holds for every $t$ such that $E[e^{itX}]\neq 0.$ I can also show that, if $E[e^{itX}]$ vanishes for countably many $t$'s, then $Y$ and $Z$ must have the same distribution. But what happens if $E[e^{itX}]$ vanishes at uncountably many points?