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
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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analysis
measure-theory
lp-spaces
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2What if $g=f$?${}$ – 2012-12-08
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0the integral is zero on an integral (say [a,b]) for every $L^p$ function g. The case f=g is not arbitrary – 2012-12-08
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1I meant to suggest that with your hypotheses, you'd have $\int f^2=0$. – 2012-12-08
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0yes but if I am not mistaken what you are saying implies f=0 a.e – 2012-12-08
1 Answers
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Yes, it implies that $f=0$ a.e. For if we let $g=f$, then we have $$\int f^2=0,$$ and $f^2\geq0$. This implies that $f^2=0$ a.e., hence $f=0$ a.e.
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0oh I see. thank you very much both of you – 2012-12-08