Rietz representation theorem for local statements of theorems I've known about Rietz representation theorem for some time, but I've always wondered if I got it right or I am missing something about local statements of theorems which make use of it.
The Riesz theorem (let me state it for $\mathbb{R}$ not to complicate things) says that the dual of $C_0 (\mathbb{R})$ is the set $\mathcal{M} (\mathbb{R})$ of finite Radon measures on $\mathbb{R}$, where $C_0$ denotes the functions vanishing at infinity. So
$$(C_0 (\mathbb{R}))^* = \mathcal{M} (\mathbb{R}) $$
The main advantage here is that for a bounded sequence of finite Radon measures (with respect to the total variation if I am not mistaken), we have weak-star compactness by standard functional analysis results. Really nice.
Now, very often it's used (implicitly) another version of the theorem, which basically says
$$ (C_c (\mathbb{R}))^* = \mathcal{M}_{loc} (\mathbb{R}),$$
where $C_c (\mathbb{R})$ denotes the set of continuous functions supported in a compact set and $\mathcal{M}_{loc}$. And so the weak-star convergence here is with respect to the functions in $C_c$.
The question is: are these two always interchangeable? I mean, the first veriosn provides a sort of more "global" behavior, the second one a completely local one. After all, the restriction of a locally finite Radon measure to a compact set is a finite Radon measure. I am asking this because many times in various courses there is a theorem and in the book it talks about finite Radon measures, while in class it is proved for locally finite Radon measures (one example that comes to my mind now are, Reshetnyak semicontinuity/continuity theorems).
I know it's not a big deal but I want to be sure once and for all. Is it really true that one can take a theorem which uses finite Radon measures weak-star converging (with respect to functions in $C_0$) to something and simply change the statement by taking locally Radon measures (and weak-star convergence with respect to $C_c$) to have a local version of the theorem? It sounds  quite right and it honestly sounds almost trivial, but the devil is in the details and maybe I'm missing something.
 A: To put it in perspective,  it is useful to look a little closer at  $C_c({\mathbb R})$,  and to note that the most natural
topology on it is the inductive limit topology.
This is defined as follows:    for every $n\in {\mathbb N}$,  consider the subspace  $E_n\subseteq C_c({\mathbb R})$ formed by the functions vanishing
outside the interval $(-n, n)$.  Equipped with the sup norm, $E_n$ is a Banach space (incidentally isomorphic to
$C_0(-n,n)$).
One can then consider the finest topology on $C_c({\mathbb R})$ among those making the natural inclusions
$$
  E_n\hookrightarrow C_c({\mathbb R})
  $$
continuous.  That topology is called the inductive limit topology and it makes $C_c({\mathbb R})$ a topological vector space, even though it may not be defined in terms of a norm.
The version of the Riesz Representation Theorem for $C_c({\mathbb R})$ mentioned by the OP corresponds to saying that the space of
continuous linear functionals on $C_c({\mathbb R})$ for the inductive limit topology
is canonically isomorphic to $\mathcal M_{{loc}}({\mathbb R})$.   This is therefore the apropriate interpretation of the expression
$$
  (C_c({\mathbb R}))^*= \mathcal M_{{loc}}({\mathbb R}).
  $$
On the other hand,  when we say that   $(C_0({\mathbb R}))^*= \mathcal M({\mathbb R})$,  we are of course referring to the continuous linear
functionals on $C_0({\mathbb R})$ for the sup norm.
Furthermore,   notice that the inclusion
$$
  \iota : C_c({\mathbb R})\hookrightarrow C_0({\mathbb R})
  $$
is continuous for the natural topologies so far referred to,  so the dual restriction map
$$
  \iota ^*: (C_0({\mathbb R}))^*\hookrightarrow (C_c({\mathbb R}))^*
  $$
becomes the inclusion
$$
  \mathcal M({\mathbb R}) \hookrightarrow   \mathcal M_{{loc}}({\mathbb R}),
  $$
since,  after all,  every finite measure is locally finite.
An analogy, which only takes Banach spaces into account, is as follows.  Given $1\leq p<q<\infty $, it is well known that every
$p$-summable sequence of scalars
is $q$-summable.  So we have an inclusion
$$
  \iota :\ell ^p\hookrightarrow\ell ^q
  $$
which is known to be continuous.  We therefore may restrict linear functionals on $\ell ^q$ to $\ell ^p$,  thus giving
rise to the dual restriction map
$$
  \iota ^*: (\ell ^q)^*\hookrightarrow (\ell ^p)^*.
  $$
Since $(\ell ^q)^*=\ell ^{q'}$ and $(\ell ^p)^*=\ell ^{p'}$, where the single quote refers to the conjugate exponent, namely
$$
  \frac 1p + \frac 1 {p'} = 1,\quad \text{and}\quad   \frac 1q + \frac 1 {q'}=1,
  $$
then   $\iota ^*$ may be viewed as a map from $\ell ^{q'}$ to $\ell ^{p'}$,  which incidentally is again the natural inclusion!
