$BV(\Omega)$ is embedded compactly in $L^1 _{\mathrm{loc}} (\Omega)$

Question

Definition: We say that a function $u: \Omega \rightarrow \mathbb{R}$ is a function of bounded variation iff $u\in L^1(\Omega)$ and $\sup\left\{\int\limits_{\Omega}u \operatorname{div}\phi : \phi \in C_c(\Omega, \mathbb{R}^d), ||\phi||_{\infty} \leq 1\right\} < +{\infty}$.

By definition it is clear that $\mathrm{BV}(\Omega)\subset L^1(\Omega)$.

How to show that this embedding is compact when $\Omega$ is bounded set? Seems like this is a very standard result, but I could not find the proof of this in many of the functional analysis book.

Answer 1

This is Theorem 5.3.4 in W.P.Ziemer: Weakly Differentiable Functions or Theorem 5.2.3 in Evans, Gariepy: Measure Theory and Fine Properties of Functions.