= SBV compactness theorem
{c}
A standard sufficient form uses a bounded <Lipschitz domain> $\Omega$, scalar functions $u_k\in SBV(\Omega)$, a uniform <essential supremum> bound, and some $p>1$ such that
$$
\sup_k\left[\int_\Omega|\nabla u_k|^p\,dx+\mathcal H^{n-1}(J_{u_k})\right]<\infty.
$$
There is a subsequence converging in <L1 norm> to a function $u\in SBV(\Omega)$. Its approximate <gradients> converge weakly in $L^p$, and the jump-set measure is bounded by the lower limit of the approximants' jump-set measures. The theorem keeps the <Cantor part of a BV derivative> out of the limit. <A Note on the Theory of SBV Functions by Giovanni Alberti and Carlo Mantegazza>[Alberti and Mantegazza, A Note on the Theory of SBV Functions, Theorem1.4; the displayed bounds are a standard sufficient special case of its uniform BV, equiintegrability and jump-control hypotheses]{ref}
The hypotheses do distinct jobs. The value bound and jump-size bound control jump variation; $p>1$ gives uniform integrability of the absolutely continuous derivatives. Ordinary <bounded-variation compactness> alone permits a diffuse singular part in the limit. Domain or tightness control also matters for global convergence: translating a fixed interval indicator arbitrarily far along $\mathbb R$ keeps all these local size bounds but gives no strongly convergent global $L^1$ subsequence.
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