$\textbf{Exercise 4.5}$ If $\mu$ is a Radon measure $g \in L^1_{loc}(\mathbb{R}^n,\mu)$ and $\int_{\mathbb{R}^n} ug d\mu \geq 0$ for every $u \in C^0_c(\mathbb{R}^n)$ with $u \geq 0$, then $g(x) \geq 0$ at $\mu$-a.e. $x \in \mathbb{R}^n$.
This is what I thought:
Suppose by contradiction that exists a subset $A \subset \mathbb{R}^n$ with $\mu(A) > 0$ and $g(x) < 0$ for every $x \in A$. I want suppose WLOG that $A \cap \text{supp} \ u \neq \emptyset$ assuming that $\mu$ is invariant under translation, then suppose WLOG that $A \subset \text{supp} \ u$ because the intersection would be non-trivial, therefore
$$\int_{\mathbb{R}^n} ug d\mu = \int_A ug d\mu < 0.$$
This contradicts the hypothesis $\int_{\mathbb{R}^n} ug d\mu \geq 0$. The problem is just know if it's true that every Radon measure on $\mathbb{R}^n$ is invariant under translation. I would like to know if this is true and a hint for this exercise if it is not true.
$\textbf{EDIT:}$
Following the hint in the comments, suppose WLOG that $A \cap \text{supp} \ u \neq \emptyset$ translating $u$ and applying the change of variables' theorem if necessary,then suppose WLOG that $A \subset \text{supp} \ u$ because the intersection is non-trivial, therefore
$$\int_{\mathbb{R}^n} ug d\mu = \int_A ug d\mu < 0.$$
This contradicts the hypothesis $\int_{\mathbb{R}^n} ug d\mu \geq 0$. $\square$