For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
Show that if $\varphi \in \mathcal{C}_{c}(\mathbb{R})$ is a positive function, we have $\|T_{\mu}\varphi\|_{1} = \|\varphi\|_{1}$.