I have some doubts reading Measure Theory and Fine Properties of Functions by Evans and Gariepy. In particular, they define the space of locally bounded variation functions $BV_{loc}(U)$ in $U\subset \mathbb R^n$, $U$ open, as the space of functions such that:
$f\in L^1_{loc}(U)$;
for all open $V\subset \subset U$ we have $\sup\{\int_V f\text{div} \phi dx:\phi\in C^1_c(V,\mathbb R^m),|\phi|\leq 1\}=C(V)<+\infty.$
Notation: $V\subset\subset U$ means: $\overline V$ is compact and $\overline{V}\subset U.$
Now, a structure theorem for functions of locally bounded variation holds (Theorem 1 pag. 167). Indeed, if $f\in BV_{loc}(U)$, the linear functional $L\colon\phi\mapsto \int_U f\text{div}\phi dx$ is "bounded" in the sense 2. above, so, by Riesz's representation theorem, one gets the existence of a Radon measure $\mu$ on $U$ and a $\mu$-measurable function $\sigma\colon U\to \mathbb R^m$ such that $|\sigma|=1, \mu$-a.e. and $$\int_U f \text{div}\phi dx=-\int_U\phi\cdot \sigma d\mu$$ for all $\phi\in C^1_c(U,\mathbb R^m).$
Now, I know that the measure $\mu$ is given by Riesz's representation theorem itself in the following way: at first, $\mu$ is defined on the open sets $O\subset U$ like this: $$\mu(O)=\sup\{\int_O f\text{div} \phi dx:\phi\in C^1_c(O,\mathbb R^m),|\phi|\leq 1\}.$$ Then, on all other subsets $B\subset U$, we put $$\mu(B)=\inf\{\mu(O):B\subset O, \,O \text{ open}\}.$$
What I don't understand is why, at page 170 of the book, there's written that $$\mu(V)=\sup\{\int_V f\text{div} \phi dx:\phi\in C^1_c(V,\mathbb R^m),|\phi|\leq 1\}$$ for all $V\subset \subset U$, without worrying that $V$ is open or not, but concerning only on the boundedness of $V$.
Can anyone help me figuring out this?