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Let $f:[a,b]\to \mathbb R$ be a measurable function .Then

Prove that if $\int _c ^d f(x)\operatorname {dx}=0$ for all $a\le c <d\le b$ then $f=0 \operatorname{a.e.}$

My try: Let $A=\{x:f(x)\neq 0\}$ .Define $A^+=\{x:f(x)>0\} $ and $A^{-}=\{x:f(x)<0\}$

We consider only the set $A^+=\{x:f(x)>0\} $ for $A^-$ we consider $-f$. Surely $A^{+}$ is a measurable set.,

How should I proceed from here?

Please give some hints

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  • Possibly duplicate http://math.stackexchange.com/questions/1385753/let-m-be-lebesgue-measure-and-a-in-r-suppose-that-f-r-to-r-is-integra?rq=1 –  Apr 28 '16 at 15:54
  • I doubt it could work effectively, because the interval structure is so simply-structured and your $A^\pm$ could be much more complicatedly structured. – Vim Apr 28 '16 at 16:24
  • @NNN;there the function is integrable but here I have measurable only – Learnmore Apr 28 '16 at 16:44

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