Showing that the convolution of two Borel measures is a Borel measure

Question

A book I am reading (Mattila's Fourier Analysis and Hausdorff Dimension) defines the convolution of two Borel measures $\mu, \nu$ over $\mathbb{R}^n$ sort of implicitly as $\int_{\mathbb{R}^n}\varphi(x)d(\mu\ast\nu)(x) = \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\varphi(x + y)d\mu(x)d\nu(y)$ for any continuous non-negative function $\varphi$ with a compact support, $\varphi\in C_0^+(\mathbb{R}^n,\mathbb{R})$. I know that this might be completely trivial question, but if that is the definition of the convolution between two Borel measures, how do we then show that $\mu\ast\nu$ is itself a measure (or a Borel measure)? Can we approximate the indicator function of any Borel set by a continuous non-negative function with a compact support? Or is there some good-to-know result which gives this immediately?