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Let $f$ be Lebesgue measurable, and $\int_a^b x^\alpha f(x) = 0$ for every $\alpha\ge 0$.

How do I show that $f(x)=0 ~ a.e.$

and if the condition change "$\alpha\ge 0$" to "$\alpha\ge k$ for some $k\ge 1$", if the statement is also true?

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    Shouldn't there be a $dx$ or a $df(x)$ in your integral?2012-12-16

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