= Structure theorem for functions of bounded variation
{title2=$Du\in\mathcal M(\Omega;\mathbb R^n),\quad|Du|(\Omega)=\operatorname{TV}(u)$}
The functional $\varphi\mapsto-\int u\operatorname{div}\varphi$ is bounded in the uniform <norm> exactly when $u$ belongs to the <BV space>. Extend it from compactly supported smooth tests to $C_0$, then apply the <Riesz-Markov-Kakutani representation theorem> to obtain the unique finite <vector Radon measure> $Du$. Its dual <norm> is the defining variation supremum. This representation should be distinguished from the deeper rectifiability results used in the <decomposition of a BV derivative>.
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