Let
- $X$ be a topological space,
- $\mathcal M(X)$ the space of regular complex Borel measures on $X$.
- $E :=C_0 (X)$ the space of $\mathbb C$-valued continuous functions on $X$ vanishing at infinity.
- $E^*$ the continuous dual of $E$. Then $\Lambda \in E^*$ is complex-valued and $\mathbb{C}$-linear.
We endow $\mathcal M(X)$ with the total variation norm $[\cdot]$ and $E$ with the supremum norm $\|\cdot\|_\infty$. Then we have
Riesz–Markov–Kakutani theorem Let $X$ be locally compact Hausdorff and $\Lambda \in E^*$. Then there exists a unique regular Borel complex measure $\mu$ on $X$ such that $$ \Lambda f=\int_{X} f \mathrm{~d} \mu \quad \text {for every} \quad f \in E \quad (\star). $$ Moreover, we have $\|\Lambda\|_{E^*} = [\mu]$. On the other hand, every $\mu \in \mathcal{M}(X)$ gives rise to an element $\Lambda \in E^*$ via formula $(\star)$. Consequently, the map $\Lambda \mapsto \mu$ is an isometric isomorphism $$ E^* \cong \mathcal{M}(X). $$
Let $\sigma (\mathcal M(X), E)$ be the weak$^*$ topology on $\mathcal M(X)$ induced by the set $E$ of test functions. Could you confirm if my below understanding is correct?
- $\Lambda \mapsto \mu$ is an isometric isomorphism between $(E^*, \|\cdot\|_{E^*})$ and $(\mathcal M(X), [\cdot])$, so it is a homeomorphism between norm topologies.
- It follows from $(\star)$ that $\Lambda \mapsto \mu$ is a homeomorphism between $\sigma(E^*, E)$ and $\sigma (\mathcal M(X), E)$.
Thank you so much!
