Compute the total variation of a linear bounded functional

Question

Let $\mu$ be a Radon measure in $\mathbb R^n$, if $L:C_C^{0}(\mathbb R^n , \mathbb R^m) \to \mathbb R$ be a linear functional, the total variation of $L$ is defined by $$|L|(A)= \sup \{ L(\varphi): \varphi \in C_C^{0}(A, \mathbb R^m), ||g|| \leq 1 \}$$ for open sets, and for $E \subset \mathbb R^n$ arbitrary $$|L|(E)=\inf \{|L|(A) : E \subset A, A \hbox { open } \}$$ if $L$ is bounded then $|L|$ is a radon measure, now let $f \in L_{loc}^{1}(\mathbb R^n , \mathbb R^m)$ ($\int_{K} ||f|| < \infty, $ for all $K$ compact) and define $f\mu : C_C^{0}(\mathbb R^n , \mathbb R^m) \to \mathbb R$ by $$f\mu(\varphi)=\int_{\mathbb R^n} \langle f | \varphi \rangle d\mu$$ show that $|f\mu|=||f|| \mu$ where $$||f||\mu(A)=\int_{A} ||f|| d\mu$$ i show that $$|f \mu|\leq ||f|| \mu$$ but for the another inequality i have problems, i try to use the density of $C_C^{0}(\mathbb R^n)$ in $L^{1}$ but i not sure if these is the right way.
Any hint or help i willl be very grateful