I need a reference. In my demonstration there is this passage:
$$\int_\Omega v(a-b)d\Omega = 0 \qquad \forall v \in V$$
With $a,b \in V$, a space that allows the integration (ex. $L^2(\Omega)$)
Where can I find a theorem that allows me to say that if this is true $\forall v \in V$, I can say that $a=b$ (except for a null set)?
The theorem you seek is the fundamental lemma of calculus of variations ( http://en.wikipedia.org/wiki/Fundamental_lemma_of_calculus_of_variations). Additionally, you need to exploit that $C_c^\infty(\Omega)\subset L^2(\Omega)$ is dense. This implies the statement.