Weak Compactness Theorem for Borel Measures Weak convergence is defined as follows:

The sequence $\{\mu_j\}$ of Borel measures on $\Bbb R^n$ converges weakly to a Borel measure $\mu$ if for all $f\in C_0(\Bbb R^n)$, $$\int f d\mu_j \to \int f d\mu$$

After this, they state a certain weak compactness theorem, which is supposed to follow from the separability of $C_0(\Bbb R^n)$.

Theorem. Any sequence $\{\mu_j\}$ of Borel measures on $\Bbb R^n$ satisfying $$\sup_j \mu_j(\Bbb R^n) < \infty$$ has a weakly converging subsequence.

I'm trying to prove the above theorem. Suppose $\{\mu_j\}$ is a sequence of Borel measures on $\Bbb R^n$ satisfying $\sup_j \mu_j(\Bbb R^n) < \infty$. We must find a weakly convergent subsequence $\{\mu_{j_k}\}$, i.e., $$\int fd\mu_{j_k} \xrightarrow{k\to\infty} \int f d\mu$$
for some Borel measure $\mu$, and all $f\in C_0(\Bbb R^n)$. I don't have much clue where to begin; could I please get any suggestions? Thanks a lot!

Reference: Fourier Analysis and Hausdorff Dimension by Pertti Mattila.
 A: What you are trying to prove is a simpler version of the Banach-Alaoglu Theorem.
In this case, because $C_0(\mathbb R^n)$ is separable and you have a sequence $\{\mu_j\}$, a slightly more direct argument can be made.
Let $\{f_k\}$ be a dense sequence in $C_0(\mathbb R^n)$. The sequence of numbers $$\Big\{\int f_1\,d\mu_j\Big\}$$
is bounded, and so it admits a convergent subsequence by Heine-Borel. That is, there is a subsequence $\{\mu_{1,m}\}$ such that
$$\Big\{\int f_1\,d\mu_{1,m}\Big\}$$ converges. Next we repeat the argument to obtain a subsequence $\{\mu_{2,m}\}$ of $\{\mu_{1,m}\}$ such that
$$\Big\{\int f_2\,d\mu_{2,m}\Big\}$$ converges. Inductively we construct a subsequence $\{\mu_{k,m}\}$ of $\{\mu_{k-1,m}\}$ such that
$$\Big\{\int f_k\,d\mu_{k,m}\Big\}_m$$ converges. Now one can show that
$$
\lim_k\int f_j\, d\mu_{k,k}
$$
exists for all $j$. Next, the density of $\{f_k\}$ implies that
$$
\varphi(f)=\lim_k\int f\,d\mu_{k,k}
$$
exists for all $f$ and it is a positive linear functional. By Riesz-Markov there exists $\mu$ such that
$$
\int f\,d\mu=\lim_k\int f\,d\mu_{k,k}
$$
for all $f\in C_0(\mathbb R^n)$.
