I'm reading the chapter 5 of the book "Measure Theory and Fine Properties" and there's one thing I don't understand. Theorem 1 (page 167) goes as follows:
The proof uses the Riesz Representation Theorem (page 49):
As you can see here, there some differences since $\sigma$ is defined on $U$ instead of $\mathbb{R}^n$ and $\mu$ on $\mathcal{B}_{U}$ instead of $\mathcal{B}_{\mathbb{R}^n}$. I've been looking for this result (written in a more general way) on the internet but I couldn't find it. Does someone know where can I find it?
Furthermore, Riesz Representation Theorem doesn't guarantee that $\sigma$ and $\mu$ are unique (if you also see the prove, it is not mentioned). Why I'm saying that? Because after Structure Theorem for $BV_{\text{loc}}$ functions they try to use $\mu$ and $\sigma$ to talk about the "derivative" of $f$ on some sense:
It's a bit confusing talking about the "derivative" of $f$ and not being unique in a certain way (like Sobolev functions). That's why I need a more general result of the Riesz Representation Theorem.
They also use the Lebesgue Decomposition Theorem which states the following:
Let $(X,\Sigma,\mu)$ a $\sigma$-finite measure space and $\nu$ a real (or complex) measure (it doesn't take the values $+\infty$ or $-\infty$). Then, there exists a unique pair of real (or complex) measures $\nu_{\text{abs}}$ and $\nu_{\text{sing}}$ over $(X,\Sigma)$ such that $\nu=\nu_{\text{abs}}+\nu_{\text{sing}}$, $\nu_{\text{abs}}\ll\mu$ and $\nu_{\text{sing}}\perp\mu$.
But, what we know about the measure with density $\sigma_i$ with respect to $\mu$ is that is a signed measure (not real, we don't know if it takes one of the values $+\infty$ or $-\infty$. I'm not sure how to prove that decomposition makes sense.
The vector-valued Riesz representation theorem holds in more generality. You can replace $\mathbb{R}^n$ by a Hausdorff locally compact topological space $X$, see Theorem 4.14 in Leon Simon's Introduction to Geometric Measure Theory. The couple $(\mu, \sigma)$ is unique, but unfortunately, this is proved neither in Evan's nor in Simon's book, so here is a quick argument. You can prove that
$$ \mu(U) = \sup \left\{ L(f) : f \in \mathscr{C}_c(X), \|f\|_\infty \leq 1, \operatorname{supp} f \subset U \right\} $$
for any open set $U \subset X$ (for example, use Theorem 4.10 in Simon's book and inner regularity with respect to compact sets). Since a Radon measure is outer regular, that is, for any $A \subset X$
$$ \mu(A) = \inf \{ \mu(U) : U \text{ open} \supset A\} $$
this shows that $\mu$ is unique. From there, it is pretty standard to show that the vector field $\sigma$ is unique ($\mu$-almost everywhere). You can use Theorem 4.10 again.