# Finite signed measures: reconcile different types of convergence

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.

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