How to generalize the integral invariant property: $\int_{\mathbb R} e^{-r^2}dr = \int_{\mathbb R} e^{-(r+t)^2}dr $

Question

The following equation holds for all $t \in \mathbb R$. $$\int_{\mathbb R} e^{-r^2}dr = \int_{\mathbb R} e^{-(r+t)^2}dr $$ I'm trying to extend the invariant property to a general unbounded set $\mathcal R$, on which a measure $\mu$ is defined, such as $\mathcal R$ is the set of all the $m*n$ column orthonormal matrices. The corresponding equation is shown in the following. $$ \int_{\mathcal R} e^{-f(r)^2}\mu(dr) = \int_{\mathcal R} e^{-(f(r)+t)^2}\mu(dr)$$ My question is: what properties of the real valued function $f(r)$, the measure $\mu$, and the set $\mathcal R$ should have in order to have the above equation for all $t\in\mathbb R$.