Riesz representation theorem if field is $\Bbb{R}$ Consider the following fragments from Folland's book:

This seems to suggest that we are working over $\Bbb{C}$. Do we still have $M(X) \cong C_0(X)$ when we consider the vector spaces over $\Bbb{R}$?
 A: Maybe, just to give a more structured answer. What is well-known is that the map
$$ M(X) \to C_0(X)^\ast, \quad \mu \mapsto I_\mu$$
is an isometric isomorphism (and this is usually refered to as the Riesz representation therorem). However, aside from the obvious Banach space structure there is also a lattice structure that can be investigated.
The space $M(X)$ is a Banach lattice with positive cone given by $M(X)_+ = \{ \mu \in M(X) : \mu \geq 0 \}$, where $\mu \geq 0$ if $\mu(A) \geq 0$ for all measurable sets $A \subseteq X$. On the other hand, $C_0(X)$ is also a Banach lattice with positive cone given by $C_0(X)_+ = \{f \in C_0(X) : f \geq 0 \}$. Hence, so is its dual $C_0(X)'$, with positive cone $C_0(X)'_+ = \{\varphi \in C_0'(X) : \varphi \geq 0 \}$, where $\varphi \geq 0$ if $\varphi(f) \geq 0$ for all $f \in C_0(X)_+$.
Now your question basicly boils down to the question if $\mu \geq 0$ if and only if $I_\mu \geq 0$ since $\mu = \mu^+ - \mu^-$ for all real measures $\mu \in M(X)$, where $\mu^+ = \mu \vee 0$ and $\mu^- = -\mu \vee 0$. This is actually the case, see here for example. There are several different Riesz representation theorems of is lattice theoretic kind, see Chapter 14, Riesz Representation Theorems in "Infinite Dimensional
Analysis - A Hitchhiker’s Guide" of Aliprantis and Border for example.
Finally, it is also worthwhile to state that you can perform all proofs on the real version of $M(X)$ and $C_0(X)'$ with just the same arguments. So you obtain all results for real Banach lattices. Then the complex theorems show in a sense that the complexifications of $M(X; \mathbb R)$ and $C_0(X; \mathbb R)'$ are given by $M(X)$ and $C_0(X)'$ and that the isometry $I$ extends nicely to these complex spaces.
I hope things got more clear :-)
