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

Question

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

Answer 1

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.