Integral result, looking for proof or reference

Question

I am looking for a proof or reference to a text which has the following result:

Let $\Omega$ be an open subset of $\mathbb{R}^{n}$. Then if $f$ is a measurable function of some sort then if it follows that for any $g \in C^{\infty}_{o}(\Omega)$ we have $$\int_{\Omega}fg dx = 0$$

then that implies that $f = 0 \text{ almost everywhere }$.

and if $f$ is continuous and for any $g \in C^{\infty}_{o}(\Omega)$ we have $$\int_{\Omega}fg dx = 0$$ then $f = 0 \text{ everywhere}.$

Thanks for any assistance. I have seen a result like this before, I just can't recall where.

Answer 1

you are looking for a theorem of calculus of variations. It is the du Bois-Reymond Lemma. There are other similar theorems, like a Lagrange theorem, that involves also derivatives.