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We characterize all (signed) measures in$BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$, where $BV_{\frac{n}{n-1}}(\mathbb{R}^n)$is defined as the space of all functions $u$ in$L^{\frac{n}{n-1}}(\mathbb{R}^n)$ such that $Du$ is a finite vector-valuedmeasure. We also show that $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$ and$BV(\mathbb{R}^n)^*$ are isometrically isomorphic, where $BV(\mathbb{R}^n)$ isdefined as the space of all functions $u$ in $L^{1}(\mathbb{R}^n)$ such that$Du$ is a finite vector-valued measure. As a consequence of ourcharacterizations, an old issue raised in Meyers-Ziemer [MZ] is resolved byconstructing a locally integrable function $f$ such that $f$ belongs to$BV(\mathbb{R}^n)^{*}$ but $|f|$ does not. Moreover, we show that the measuresin $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$ coincide with the measures in $\dotW^{1,1}(\mathbb{R}^n)^*$, the dual of the homogeneous Sobolev space $\dotW^{1,1}(\mathbb{R}^n)$, in the sense of isometric isomorphism. For a boundedopen set $\Omega$ with Lipschitz boundary, we characterize the measures in thedual space $BV_0(\Omega)^*$. One of the goals of this paper is to make precisethe definition of $BV_0(\Omega)$, which is the space of functions of boundedvariation with zero trace on the boundary of $\Omega$. We show that themeasures in $BV_0(\Omega)^*$ coincide with the measures in$W^{1,1}_0(\Omega)^*$. Finally, the class of finite measures in $BV(\Omega)^*$is also characterized.

Characterizations of signed measures in the dual of $BV$ and related isometric isomorphisms