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
- $$ \sigma(\mathcal M(X), \mathcal C_c(X)) \subset \sigma(\mathcal M(X), \mathcal C_0(X)) \subset \sigma(\mathcal M(X), \mathcal C_b (X)). $$
- $\sigma(\mathcal C_c(X),\mathcal M(X))$ 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
- $$ \sigma(\mathcal C_0 (X), \mathcal M(X)) = \sigma(\mathcal C_0 (X), (\mathcal C_0 (X))^*). $$
- $\sigma(\mathcal M(X), \mathcal C_0(X))$ and $\sigma((\mathcal C_0 (X))^*, \mathcal C_0(X))$ have exactly the same topological properties.