If $\int u\varphi = 0$ for all $\varphi \in C_c^\infty(M)$ with $\int_M \varphi =0$, is $u=0$ a.e.?

Question

Let $M$ be a compact Riemannian manifold and $u \in L^2(M)$. we know that if for all $\varphi \in C_c^\infty(M)$, $$\int_M \varphi u = 0,$$ then $u=0$ a.e.

Suppose $$\int_M \varphi u =0$$ for all $\varphi \in C_c^\infty(M)$ with $\int_M \varphi =0$. Then does the conclusion still hold that $u=0$ a.e?

Answer 1

No, consider $u=1\mbox{}{}{}{}{}{}{}{}{}$.

Answer 2

The answer is affirmative if we consider $C_0$ and assume $u\in L^1_{loc}$. Weak derivative being zero implies $u$ has to be zero $a.e$ see here