Finite signed measures: reconcile different types of convergence

Question

Let

• $X$ be a metric space,
• $\mathcal M(X)$ the space of all Borel signed measures on $X$,
• $\mathcal C_b(X)$ be the space of real-valued continuous functions,
• $\mathcal C_0(X)$ the space of real-valued continuous functions that vanish at infinity, and
• $\mathcal C_c(X)$ the space of real-valued continuous functions with compact supports.

I'm reading this answer about different types of convergence. Below is my understanding. Could you verify if it is fine?

---

Notice that $\mathcal M (X)$ is a real Banach space with total variation norm. Also, $\mathcal C_b(X)$ and $\mathcal C_0(X)$ are real Banach space with supremum norm. Below we use the kind of topology induced by dual pairing. Because $$ \mathcal C_c(X) \subset \mathcal C_0(X) \subset \mathcal C_b(X), $$ we have

1. $$ \sigma(\mathcal M(X), \mathcal C_c(X)) \subset \sigma(\mathcal M(X), \mathcal C_0(X)) \subset \sigma(\mathcal M(X), \mathcal C_b (X)). $$
2. $\sigma(\mathcal C_c(X),\mathcal M(X))$ and $\sigma(\mathcal C_0(X),\mathcal M(X))$ are the subspace topologies that $\sigma(\mathcal M(X), \mathcal C_b (X))$ induce on $\mathcal C_c(X), \mathcal C_0(X)$ respectively.

The closure of $\mathcal C_c(X)$ in $\mathcal C_b(X)$ is $\mathcal C_0(X)$, so $$ \sigma(\mathcal M(X), \mathcal C_c(X)) = \sigma(\mathcal M(X), \mathcal C_0(X)). $$

Usually, $\sigma(\mathcal M(X), \mathcal C_b(X))$ and $\sigma(\mathcal M(X), \mathcal C_0(X))$ are called the weak and weak$^*$ topologies of $\mathcal M(X)$ respectively. We have $\mathcal M (X)$ can be isometrically embedded into the continuous dual $(\mathcal C_0 (X))^*$ of $\mathcal C_0 (X)$, so $$ \sigma(\mathcal C_0 (X), \mathcal M(X)) \subset \sigma(\mathcal C_0 (X), (\mathcal C_0 (X))^*). $$

If $X$ is locally compact, then $\mathcal{M}(X)$ is isometrically isomorphic to $(\mathcal C_0 (X))^*$ by Riesz–Markov–Kakutani theorem and thus

1. $$ \sigma(\mathcal C_0 (X), \mathcal M(X)) = \sigma(\mathcal C_0 (X), (\mathcal C_0 (X))^*). $$
2. $\sigma(\mathcal M(X), \mathcal C_0(X))$ and $\sigma((\mathcal C_0 (X))^*, \mathcal C_0(X))$ have exactly the same topological properties.

Answer 1

I was wrong in saying that

The closure of $\mathcal C_c(X)$ in $\mathcal C_b(X)$ is $\mathcal C_0(X)$, so $$ \sigma(\mathcal M(X), \mathcal C_c(X)) = \sigma(\mathcal M(X), \mathcal C_0(X)). $$

In fact, if the mass escapes at infinity, then above statement is false. In fact, we have below theorem from the paper Vague and weak convergence of signed measures of Martin Herdegen, Gechun Liang, and Osian Shelley, i.e.,

Proposition 1.3. Let $\Omega$ be locally compact and $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ with $\sup _{n \in \mathbb{N}}\left\|\mu_n\right\|<\infty$. Then $I_{\mu_n}(f) \rightarrow I_\mu(f)$ for all $f \in C_0(\Omega) \quad$ if and only if $\quad I_{\mu_n}(f) \rightarrow I_\mu(f)$ for all $f \in C_c(\Omega)$.