Strict convergence in BV ($u_k\to u\text{ in }{L}^1,|D{u}_k|(\Omega )\to |Du|(\Omega )$)

Strict convergence combines strong convergence in $L^1$ with convergence of the total variation seminorm. It implies weak-star convergence in BV but does not make jump sets stable: mollifications of an interior step can converge strictly to the step while every approximant has an empty jump set of a bounded-variation function.