Let $F,G$ be distributions such that $F$ is zero IFF $G$ is zero. Then there is $c\in\mathbb{C}$ such that $F=cG$.
Can I argue this proposition from the integral definition? This is to say that given a test function $\psi$ the integral $\int_{-\infty}^{+\infty}f(x)\psi(x)dx=0$ only if $\int_{-\infty}^{+\infty}g(x)\psi(x)dx=0$. (Now here can I use the same test function for both integrals?).
The problem is to determine what happens when $F$ is not zero, but can't I make it zero again by means of some change of variables (like $t=x-a$)? I expect that after some change of variables both integrals will be zero again so some constants will come out after changing variables.