It is well known that if $f$ defined on a compact interval $[0,T]$ is a continuous function with finite variation, then $f$ induces a signed measure $\mu$ on $[0,T]$. Let $|\mu|$ be the total variation measure associated to $f$, that is, $|\mu|=\mu^++\mu^-$ where $\mu=\mu^+-\mu^-$ is the Jordan decomposition of a signed measure.
My question is: do we have, for all $t\in [0,T]$, $|\mu|([0,t])=V_t(f)$, where $V_t(f)$ denotes the total variation of $f$ on $[0,t]?$