Let $\mu_1,\mu_2$ be two locally finite complex regular Borel measures on $[0,+\infty)$ and $\delta_x$ be the Dirac measure at point $x\in[0,+\infty)$. Suppose that for all $x\in(0,+\infty)$ $$\delta_x*\mu_1=\delta_x*\mu_2.$$ Could we say that $\mu_1=\mu_2$?
Where $*$ is convolution of measures which is defined for measures $\mu,\nu$ $$\int_0^\infty\psi(x)d(\mu*\nu)(x)=\int_0^\infty\int_0^\infty\psi(x+y)d\mu(x)d\nu(y), \quad \psi\in C_0([0,+\infty)).$$ In other world could we conclude that $$\lim_{n\to\infty}\delta_{\frac{1}{n}}*\mu=\delta_0*\mu$$ for all locally finite complex regular Borel measure $\mu$?