Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.