  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-valued
measure. We also show that $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$ and
$BV(\mathbb{R}^n)^*$ are isometrically isomorphic, where $BV(\mathbb{R}^n)$ is
defined 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 our
characterizations, an old issue raised in Meyers-Ziemer [MZ] is resolved by
constructing a locally integrable function $f$ such that $f$ belongs to
$BV(\mathbb{R}^n)^{*}$ but $|f|$ does not. Moreover, we show that the measures
in $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$ coincide with the measures in $\dot
W^{1,1}(\mathbb{R}^n)^*$, the dual of the homogeneous Sobolev space $\dot
W^{1,1}(\mathbb{R}^n)$, in the sense of isometric isomorphism. For a bounded
open set $\Omega$ with Lipschitz boundary, we characterize the measures in the
dual space $BV_0(\Omega)^*$. One of the goals of this paper is to make precise
the definition of $BV_0(\Omega)$, which is the space of functions of bounded
variation with zero trace on the boundary of $\Omega$. We show that the
measures 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.
