Let $\{K_n\}$ be approximating unit and $\mu\in M(\mathbb{T})$. Show that $\mu\ast K_n\to \mu$ weakly means $$\int f(t)d(\mu\ast K_n)(t)\to\int f(t)d\mu(t)$$
Suppose $\mu\in L^1(\mathbb{T})$, I know that $\mu\ast K_n\to\mu$ and we are done. but what happens when $\mu\not\in L^1(\mathbb{T})$? How can I prove that?
EDIT: I know that $K_n\in L^1$ and that $\int_\mathbb{T}K_n(s)\frac{dt}{2\pi}=1$ but I can't procceed from here.
EDIT2: Maybe to use $f\ast K_n\to f$ if $f\in L^1$?