Suppose a measurable function $|g(x)|\leq M$ is bounded above by $M>0$.
If for every interval $[a,b]\subset\mathbb{R}$ $$\int_a^b g(x)\,dx=0$$ Is the following statement true?
$$\forall f\in L^1,\int_{\mathbb{R}}f(x)g(x)dx = 0$$
I am not sure how to proceed, and thanks in advance for any ideas.
I don't see how this can work in general. Take for instance $f(x)=g(x)=\sin(x)$ on $[0,2\pi]$, then $$\int_0^{2\pi} g(x)\,dx=0$$ however $$\int_0^{2\pi} f(x)g(x)\,dx=\int_0^{2\pi}\sin^2(x)\,dx>0$$