Say that $X$ and $Y$ are independent random variables, both with exponential($\lambda=1)$ distributions. And we are dealing with the transformations $U=X-Y$ and $V=Y$.
My question is what is the range of $u$ values?
For example, we know $x>0$ and $y>0$. Does this mean that $-\infty<x-y<\infty$, and hence $-\infty<u<\infty$?
For every interval $(a,b)\subseteq \mathbb R$ where $a<b,$ there are some values of $X>0$ and $Y>0$ for which $a<X-Y<b,$ and in fact $\Pr(a<X-Y<b)>0.$ Therefore nothing smaller than the interval $(-\infty,\infty)$ can serve.