We need to show that image $f(A)$ of a rectifiable set $A$, under a diffeomorphism $f$, is also rectifiable.
Definition: A rectifiable set is a set which is closed, bounded & has boundary of measure $0$.
Approach: It is easy to show that $f(A)$ is closed & bounded. Also, I can show that a measure $0$ set under a diffeomorphism has measure $0$. I am confused if these two arguments enough. As the set $A$ has boundary of measure $0$ and the diffeomorphism $f$ would take the boundary $B$ to $B'$. And I can show measure of $B'$ is $0$. Now the problem is is it confirmed that $B'$ is the boundary of $f(A)$?