Note: I am doing this question just for fun, not for hw.
Question: Fix any dense subset $G$ of the unit ball of $C^{0}(M)$. Here $C^0(M)$ refers to the space of continuous functions defined on $M$, a metric space. Show that a sequence of probability measures $(P_n)_n$ on M converges to to some P $ \in M_1(M)$ in the weak* topology if and only if $\int \phi d P_n$ converges to $\int \phi d P$ for every $\phi \in G$.
Note: here, $M_1(M)$ refers to the set of Borel probability measures on M.
Here is the proof of a similar but different question. A sequence of probability measures $(P_n)_n$ on M converges to to some P $ \in M_1(M)$ in the weak* topology if and only if $\int \phi d P_n$ converges to $\int \phi d P$ for every bounded continuous function $\phi: M \rightarrow \mathbb{R}$.
Proof of this question: First, a quick definition of the weak* topology. Given a measure $P \in M_1(M)$, a finite set $\phi=\{\phi_1...\phi_N\}$ of bounded continuous functions $\phi_i: M \rightarrow \mathbb{R}$, and a number $\epsilon >0$, we define $V(P,\phi, \epsilon)$=$\{\nu \in M_1(M): | \int \phi_i d\nu -\int \phi_i d\mu|< \epsilon \mbox{ for every i }\}$. The family $\{V(P,\phi, \epsilon): \phi, \epsilon\}$ may be taken as a neighborhood of each $P \in M_1(M)$. The weak* topology is defined by these bases of neighborhoods.
Now for one direction of the proof. Consider any set $\phi=\{\omega\}$, where $\omega$ is a single bounded continuous function. Since $(P_n)_n \rightarrow P$, we have that for any $\epsilon>0$, there exists $N \geq 1$ such that $P_n \in V(P,\phi,\epsilon)$ for every $n\geq N$. This just means that, $|\int \omega dP_n -\int \omega dP|< \epsilon$ for every $n\geq N$. $(\int \omega dP_n)_n$ converges to $\int \omega dP $.
For the converse, assume that $(\int \omega P_n)_n$ converges to $ \int \omega P$ for every bounded continuous function $\omega$. We need to show that given $\phi=\{\omega_1,..\omega_M\}$ and $\epsilon >0$, we have that there exists $N \geq 1$ such that $P_n \in V(P, \phi, \epsilon)$ for $n \geq N$. To show this, let: $\phi=\{\omega_1,...\omega_M\}$. Our hypothesis ensures that for every $i$ there exists $N_i$ such that $|\int \omega_i dP_n-\int \omega_i dP| < \epsilon$ for every $n >N_i$. Taking N=max$\{N_1....,N_M\}$, we have that $P_n \in V(P,\phi,\epsilon)$ for every $n>N$. Hence, proved.
Thus, I know how to prove the result for every bounded continuous function $\phi: M \rightarrow \mathbb{R}$. I, however don't understand how the proof changes when we only consider $\phi \in G$ where G is a dense subset of the unit ball of $C^0(M)$ where $C^0(M)$ is the space of continuous functions defined on M. Since I proved the result for all bounded and continuous $\phi: M \rightarrow \mathbb{R}$, then doesn't the case where $\phi$ is in a dense subset of the unit ball of continuous functions defined on $M$ just a special case of the previous one and just immediately follow. It seems too trivial and I feel as if though I am missing something... Any help would be much appreciated, thank you.
As you mentioned $P_n$ converges in the weak* topology to $P$ if \begin{equation*} \int f d P_n \to \int f d P \end{equation*} for every bounded continuous function $f$ defined on the metric space $M$. We just need to verify this condition.
Fix the bounded continuous function $f$. Without loss of generality we assume $f$ is bounded uniformly by $1$, since otherwise we can replace $f$ by $f / ||f||$, where $||f|| := \sup_{x \in M} |f(x)|$. For any $\varepsilon > 0$, pick some $\phi \in G$ that approximates $f$ uniformly on $M$, i.e. $\sup_{x \in M} |f(x) - \phi(x)| < \varepsilon$. We have the estimate \begin{align*} \left| \int f d P_n - \int f d P \right| &\leq \left| \int (f-\phi) d P_n \right| + \left| \int \phi d P_n - \int \phi d P \right| + \left| \int (\phi - f) d P \right| \\ &\leq 2 \varepsilon + \left| \int \phi d P_n - \int \phi d P \right| \end{align*} Taking limit on both sides yields \begin{equation*} \limsup_{n} \left| \int f d P_n - \int f d P \right| \leq 2\varepsilon \end{equation*} Since $\varepsilon$ is arbitrary, we conclude $\int f d P_n \to \int f d P$.