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If I have an expected value of a product of two functions, $E[f(x)g(x)]=0$ for any $g(x)$, does it mean $f(x)=0$ ?

Thanks

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    What space are you working in? What restrictions do you have on $f$ and $g$?2012-04-03
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    $L^2$, and g is bounded and measurable2012-04-03
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    **Hint:** Take $g(x)=f(x)$.2012-04-03
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    Right, thanks .2012-04-03
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    What about sets of measure zero?2012-04-03
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    I guess $g(x)=1_{f>0}(x)$ and $g(x)=-1_{f<0}(x)$ becasue the choice f=g is nice but maybe f is not bounded.2012-04-04
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    @Kolmo Yes, your idea is better. The condition that $g$ is bounded was added after I wrote my comment.2012-04-05

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