Let $g : [a,b] \mapsto \mathbb{R}$ be a function, where $[a,b]$ is a compact subset of $\mathbb{R}$, one can define the total variation of $g$ on $[a,b]$ to be
$$ TV_{a}^{b}(g) = \sup_{\mathcal{T}}\sum_{i=1}^{n_T}|g(x_{i+1}) - g(x_i)|, $$
where the supremum is over all finite partition sets $\mathcal{T} = \{x_1,....,x_T\}$, for $T\in \mathbb{N}$, and $a\leq x_1<x_2<...<x_T\leq b$. Some authors give this notation $TV_{a}^{b}(g) = \int_a^b |dg(x)|$.
My question refers to the multivariate extension to this notation. NB: I have done a fair amount of research and I understand that there are multiple definitions of Total Variation in multivariate settings that lead to different kinds of generalisation.
I would like to extend the definition of $\int_a^b |dh(\boldsymbol{x})|$ for some function $h: [a,b]^d \mapsto \mathbb{R}$, for some fixed integer $d>0$. I have not been able to find any references that use $\int_a^b |dh(\boldsymbol{x})|$. If this notation exists, would this simply be the equivalent notation for the Vitali Variation (see https://www.encyclopediaofmath.org/index.php/Vitali_variation#Fr)?