In functional analysis,

Definition A:

for any normed linear space $(X, \| \cdot \| )$, the weak star topology $\sigma (X^*, X)$ on $X^*$ is generated by the collection of seminorms $\{ p_x \, | \, x \in X\}$, defined by $$p_x (f) = |f(x)|.$$

In probability theory (more specifically from the book "Probability measures on Metric Spaces" written by Parthasarathy),

Definition B:

for any metric space $X$, let $\mathcal{M} (X)$ denote the space of measures defined on $\mathcal{B} (X)$ and let $C(X)$ be the space of all bounded real valued continuous functions on $X$, equipped with the sup norm. Then the weak topology on the space $\mathcal{M} (X)$ is generated by the base of open neighbourhoods at a point $\mu$ defined by $$ \bigg\{ \nu \in \mathcal{M} (X) \, \Bigg| \, \, \bigg| \int_X f_i \, d \nu - \int_X f_i \, d \mu \bigg| < \epsilon_i, \, \, i= 1,2,\ldots, k \bigg\},$$ where $f_1, \ldots, f_k \in C(X)$ and $\epsilon_1 , \ldots, \epsilon_k >0$.

Here is what I don't understand:

If $X$ is a compact metric space, then by the representation theorem of bounded linear functionals on $C(X)^*$, then for any $\Lambda \in C(X)^*$, there exists a unique Borel measure $\mu \in \mathcal{M}(X)$ such that $$ \Lambda_\mu (f) := \Lambda (f) = \int_X f \,d \mu, \quad \forall f \in C(X),$$ and that $$\| \Lambda_\mu \| = \mu (X).$$ Thus, if we identify each element $\mu \in \mathcal{M} (X)$ by $\Lambda_\mu \in C(X)^*$, Definitions A and B are the same.

However, for any general metric space $X$ that is NOT necessarily compact, $\mathcal{M} (X)$ and $C(X)^*$ are not necessarily in isometric isomorphism. (Or is there a representation result in greater generality?)

Is Definition B slightly more general than Definition A to cater for the needs in probability theory? If this is the case, then some functional analytic results might not be applicable to weak convergence theory in probability..... Any ideas?


For any metric space $X$, we can certainly think of any finite measure as a linear functional on $C(X)$, so there is a natural map $\mathcal{M}(X) \to C(X)^*$ which is an isometry. You are right that for non-compact $X$, this map is not surjective. (The "extra" elements of $C(X)^*$ can, I believe, be characterized as regular finitely additive measures. Dunford and Schwartz have a discussion of these objects. They are typically pathological and need the axiom of choice to construct, and probabilists usually don't want to deal with them.)

The probabilist's "weak" topology on $\mathcal{M}(X)$ can be seen as the subspace topology induced by equipping $C(X)^*$ with its weak-* topology, and considering $\mathcal{M}(X)$ as a subset of it. One also commonly studies the even smaller subset $\mathcal{P}(X)$ containing only the probability measures.

You are right that some functional analytic results about the weak-* topology on $C(X)^*$ do not directly apply to $\mathcal{M}(X)$, or need different proofs.

If $X$ is non-compact but is locally compact, we can consider instead $C_0(X)$, the space of continuous functions $f : X \to\mathbb{R}$ which "vanish at infinity", i.e. the closure in $C(X)$ of the compactly supported functions $C_c(X)$. (Some authors use $C_0(X)$ for compactly supported functions, and use $C_\infty(X)$ for functions vanishing at infinity.) There is an isometry $\mathcal{M}(X) \to C_0(X)^*$ and there is a version of the Riesz representation theorem stating that it is surjective. But the weak-* topology on $C_0(X)^*$ doesn't give the probabilist's weak topology on $\mathcal{M}(X)$; in particular, $\mathcal{P}(X)$ is not weak-* closed in $C_0(X)^*$.


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