1

I'm trying to strengthen my understanding between these two topologies of $\mathcal P(E)$. Could you have a check on my below exposition?

Let

  • $X$ be a normal Hausdorff topological space,
  • $E :=C_b (X)$ the space of all real-valued bounded continuous functions on $X$,
  • $C_0 (X)$ the space of all real-valued continuous functions on $X$ that vanish at infinity,
  • $C_c (X)$ the space of all real-valued continuous functions on $X$ with compact supports,
  • $\mathcal P(X)$ the space of all Borel probability measures on $X$, and
  • $\mathcal M(X)$ the space of all finite signed Borel measures on $X$.

We endow $E$ with the supremum norm $\|\cdot\|_\infty$. Then $(E, \|\cdot\|_\infty)$ is a Banach space and $C_0 (X)$ its closed subspace. Also, $C_c (X)$ is not generally closed in $(E, \|\cdot\|_\infty)$. Let $\mu, \mu_n \in \mathcal P(X)$ for all $n \ge 1$.

  • We define the weak topology of $\mathcal P(X)$ as the one induced by the convergence (denoted by $\rightharpoonup$) criterion $$ \mu_n \rightharpoonup \mu \overset{\text{def}}{\iff} \int_X f \mathrm \mu_n \to \int_X f \mathrm \mu \quad \forall f \in E. $$

  • We define the weak$^*$ topology of $\mathcal P(X)$ as the one induced by the convergence (denoted by $\overset{\ast}{\rightharpoonup}$) criterion $$ \mu_n \overset{\ast}{\rightharpoonup} \mu \overset{\text{def}}{\iff} \int_X f \mathrm \mu_n \to \int_X f \mathrm \mu \quad \forall f \in C_0 (X). $$

  1. Because the closure of $C_c (X)$ in $(E, \|\cdot\|_\infty)$ is $C_0 (X)$, we have $$ \mu_n \overset{\ast}{\rightharpoonup} \mu \iff \int_X f \mathrm \mu_n \to \int_X f \mathrm \mu \quad \forall f \in C_c (X). $$

  2. Let $E^*$ be the continuous dual space of $E$, and $\sigma(E^*, E)$ the weak$^*$ topology of $E^*$. We endow $\mathcal M(X)$ with the total variation norm $[\cdot]$. Then $(\mathcal M(X), [\cdot])$ is a Banach space. There is an isometric embedding $$ \varphi:\mathcal M(X) \to E^*, \mu \mapsto \varphi(\mu), $$ defined by $$ \langle \varphi(\mu), f \rangle :=\int_X f \mathrm d \mu \quad \forall f \in E. $$

Then the weak topology of $\mathcal P(X)$ is the subspace topology that $\sigma(E^*, E)$ induces on $\varphi [\mathcal P(X)]$.

  1. Let $F:=C_0(X)$. Then $F$ is a Banach space. We define a map $$ \psi:\mathcal P(X) \to F^*, \mu \mapsto \psi(\,u) $$ by $$ \langle \psi(\mu), f \rangle :=\int_X f \mathrm d \mu \quad \forall f \in F. $$

Because $F$ is dense in $(E, \|\cdot\|_{L_1})$, we get $\psi$ is injective. Clearly, $\mu_n \overset{\ast}{\rightharpoonup} \mu$ if and only if $\psi(\mu_n) \to \psi(\mu)$ in the weak$^*$ topology $\sigma(F^*, F)$ of $F^*$.

Analyst
  • 5,637

0 Answers0