Let $f$ be of bounded variation on $[a,b]$, and define $ν(x)=TV(f_{[a,x]}) ∀x∈[a,b]$,
I want to show: $\int_a^b|f'|= TV(f)$ iff $f$ is absolutely continuous on $[a,b]$.
My attempts:
I've shown $|f'|\leq v'$ a.e. on $[a,b]$ and infered that $\int_a^b|f'|\leq TV(f)$.