Decomposition of functions of bounded variation in Ahlfors-regular PI spaces

We prove that, on a non-compact Ahlfors-regular PI space, every function of bounded variation $f$ can be decomposed as a countable sum ${\sum }_{n=1}^{\mathrm{\infty }}{f}_n$ of monotone functions of bounded variation $f_n$, in such a way that the total variation of $f$ is the sum of the total variations of the functions $f_n$. Our result extends a Euclidean result of Bianchini and Tonon to the metric setting.