Is this equivalent to bounded variation?

Question

Let $\Omega \subset \mathbb{R}^n$ be open and bounded with smooth boundary. For $f \in L^1(\Omega)$, define $$ \|D_1 f\|_M(\Omega) =\inf\left\{\liminf_{k\to\infty}\int_\Omega |\nabla f_k|\,dx \mid f_k \to f \text{ in } L^1(\Omega),\ f_k \in \text{ Lip }(\Omega)\right\}. $$ Here $\text{Lip}(\Omega)$ is the set of Lipschitz functions on $\Omega$. Note that by Rademacher's Theorem, for $f \in \text{Lip}(\Omega)$, $\nabla f$ exists Lebesgue-a.e. My question is, is $\|D_1 f\|_M(\Omega)$ the same as $\int_\Omega |Df|$ in general? I have a feeling the answer is ''no'', because if it is ''yes'', people would probably use this as the definition of bounded variation instead of the usual definition, which I find more complicated.

Answer 1

Yes, this is another way to introduce the BV norm, sometimes called Miranda's definition. People do use it, but it does not mean the distributional definition can be forgotten. It's not a bad thing to have two or more definition of the same class. For example, they might generalize in different ways when we move beyond Euclidean spaces. This dissertation is relevant.