If I have values $v$ uniformly distributed over $\left[1, 2\right]$ where $\operatorname{F}\left(v\right) = v - 1$, is this equivalent to a uniform distribution where $\operatorname{F}\left(v\right) = v$ over $\left[0, 1\right]\ ?$.
It seems to me that it should be but I want to make sure that I am not missing any subtle points.
You have distributions $F_1(v)=(v-1)\mathbf 1_{v\in[1..2]}$ and $F_2(v)=(v)\mathbf 1_{v\in[0..1]}$, which are clearly not equivalent.
However, since $F_1(v)=F_2(v-1)$ we can say that these distributions are related by a linear shift.