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If $f$ is an $L^p$ function and $\int f(x)g(x)dx=0$ for every $L^p$ function $g$ does that imply that $f=0$ a.e

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    What if $g=f$?${}$2012-12-08
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    the integral is zero on an integral (say [a,b]) for every $L^p$ function g. The case f=g is not arbitrary2012-12-08
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    I meant to suggest that with your hypotheses, you'd have $\int f^2=0$.2012-12-08
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    yes but if I am not mistaken what you are saying implies f=0 a.e2012-12-08

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