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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 $\int_a^b|f'|\leq TV(f)$.

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What's your definition of $\int_a^b|f'|$, explicitly in terms of say supremums or infimums or what-not? – Alex R. Dec 4 '12 at 23:15
The notes at the end of Chapter 20 "Differentiation" in Neal L. Carothers Real Analysis refer to Wheeden, R. L. and Zygmund, A. [1977] Measure and Integral for a proof that $v'(x) = |f'(x)|$ a.e. when $f \in BV[a,b]$. – kahen Dec 4 '12 at 23:17

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