What is the exact connection between BV-Functions and their weak derivatives

Question

let $ \Omega \subseteq \mathbb{R}^n$ be open. I have read that $u : \Omega \rightarrow \mathbb{R}$ is of bounded Variation if there exist a Radon measure $\mu : \mathcal{B}(\Omega) \rightarrow \mathbb{R}^n$ with $\displaystyle \int_\Omega u div\varphi d\mathcal{L}^n = \displaystyle\int_\Omega \langle\varphi, \mu \rangle, \forall \varphi \in C_c^1(\Omega,\mathbb{R}^n)$, with $\mathcal{L}^n$ denoting the n-dimensional Lebesgue-measure. And $\mu$ is exactly the measure corresponding to the weak derivative or the measure corresponding to the distributional derivative? I'm not entirely sure what the difference between these 2 is. And what does corresponding mean in this context?