For positive integer $k$, let $\mu_k=\dfrac{1}{2}\left(\delta(x)+\delta\left(x-\dfrac{2}{3^k}\right)\right)$. Let $dC_k=\mu_1\ast\cdots\ast\mu_k$. In this question, it was shown that $$\lim_{k\rightarrow\infty}dC_k((-\infty,x))=C(x),$$ where $C$ is the Cantor function.
Show that $dC_k$ converges weakly to $\mu_C$.
By definition of weak convergence, we need to show that $$\lim_{k\rightarrow\infty}\int_\mathbb{R}fdC_k=\int_\mathbb{R}fd\mu_C$$ for any function $f\in C_0^\infty(\mathbb{R})$. But it's not clear how to continue by using the result above.
