Let $M$ be a locally compact Hausdorff space. A continuous real valued function $f \colon M \to \mathbb R$ is said to vanish at infinity if, for every $\epsilon > 0$, there exists a compact set $K \subset M$ such that $$\sup_{x\in M-K} |f(x)| < \epsilon$$ Denote by $C_0(M)$ the space of all continuous functions $f \colon M \to \mathbb R$ that vanish at infinity (see Exercise 3.2.10).
(a) Prove that $C_0(M)$ is a Banach space with the supremum norm.
no problem here
''(b)The dual space $C_0(M)^∗$ can be identified with the space $\mathcal{M}(M)$ of signed Radon measures on M with the norm (1.1.4) [They refer to the total variation to be the norm], by the Riesz Representation Theorem (see [75, Thm. 3.15 & Ex. 3.35]). Here a signed Radon measure on $M$ is a signed Borel measure μ with the property that, for each Borel set $B \subset M$ and each $\epsilon$ >0, there exists a compact set $K \subset B$ such that |$\mu(A)−\mu(A \cap K)| <\epsilon$ for every Borel set $A \subset B$.''
What exactly am I being asked to do here, other than quote the result of Riesz? I'm very confused here. What exactly does it mean to prove spaces ''can be identified'' with another?
(c) Prove that the map $\delta \colon M \to C_0(M)^∗$, which assigns to each $x \in M$ the bounded linear functional $\delta_x \colon C_0(M) \to \mathbb R$ given by $\delta_x(f) := f(x)$ for $f \in C_0(M)$, is a homeomorphism onto its image $\delta(M) \subset C_0({M})^∗$, equipped with the weak$^*$ topology. Under the identification in (b) this image is contained in the set $P(M) := \{\mu \in \mathcal{M}(M):
\mu \ge 0, \|\mu\| = \mu(M) = 1\}$ of Radon probability measures. Determine the weak$^*$ closure of the set $\delta(M) = \{ \delta_x | x \in M \} \subset$ $P(M)$.
Best Answer
I think part (b) is not really meant as a question, just as information that you will need to solve part (c). However, you might take it as an opportunity to solve Exercise 5.35 (not 3.35, it's a typo) of the book [75], which is Measure and Integration by Dietmar A. Salamon. Note that the statement at hand is not exactly what is shown by [75] Theorem 3.15, which is only about positive measures and positive linear functionals.
Formally, "can be identified with" would be to say that there is an isometric isomorphism between these two normed spaces: a bijective linear map $T : \mathcal{M}(M) \to C_0(M)^*$ which is an isometry (i.e. $\|T\mu\|_{C_0(M)^*} = \|\mu\|_{TV}$). The map should be understood to be $(T\mu)(f) = \int f\,d\mu$. Proving that this map is a bijection and an isometry are parts (iii) and (i) of [75] Exercise 5.35. So once this has been shown, you can think of the measure $\mu$ and the functional $T\mu$ as "the same object", and whenever you are asked to prove something about a measure, you could instead prove the corresponding fact about the corresponding functional, or vice versa.
In particular, the sentence "under the identification in (b)" in part (c) should be understood as saying that $T^{-1}(\delta(M)) \subset P(M)$.