I have a question about this book, "Topics in Random Matrices Theory" of Terence Tao.
He claims that if $$\int_{\mathbb{R}}\varphi \ d\mu_{\frac{1}{\sqrt{n}}M_n}\stackrel{P}\rightarrow\int_{\mathbb{R}}\varphi \ d\mu\hspace{2cm}(1)$$ for every $\varphi\in C_b(\mathbb{R})$, i.e, for every function $\varphi$ continuous and bounded, then $$\hspace{1cm}\mu_{\frac{1}{\sqrt{n}}M_n}\stackrel{P}\rightarrow\mu\hspace{3.4cm}(2)$$
This looks natural, and in fact, he doesn't even prove this, looks like it is an obvious result. First I thought it would follow directly from the definition of weak convergence, but the convergence in (1) is weaker then weak convergence, and I could'n find any result about this implication. I hope someone help me on this?
New Add: Here is another related text talking about the same thing. Maybe someone could read this and explain to me what is really going on.
Thank you very much.