Suppose $f:\mathbb{R}\to\mathbb{R}$ is Borel measurable. I want to show that $\displaystyle \int_a^b f(x)\mathrm dx=0$ for all $-\infty$f=0$ a.e. I'm really not sure how to approach this problem; any help would be welcome.
$\int_a^b f(x)\mathrm dx=0$ for all $-\infty a,b rational implies $f=0$ a.e
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real-analysis
measure-theory
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4Consider the collection C of Borel sets B such that the integral of f on B is zero. You already know C contains every interval (a,b). What is the structure of C and what theorem from your course does this evoke? – 2011-08-24
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0Perhaps the Lebesgue Density Theorem? – 2011-08-24