Let $G$ be a non-discrete locally compact group with left Haar measure $\mu$.
There is an isometric embedding of $L^{1}(G)\to M(G), f\mapsto fd\mu$.
Since $G$ is not discrete, the point-mass measure $\delta_{x}$ is not in $M_{a}(G)$, but it is still a non-zero element of $M(G)$, and so can be convoluted with elements of $L^{1}(G)$.
Taking $(e_{\alpha})$ to be an approximate identity for $L^{1}(G)$.
Does $\displaystyle\lim\limits_{\alpha}\delta_{x}*e_{\alpha}$ exist in $M(G)$?