Compactness and lower semicontinuity for bounded-domain BV

For $u_k\to u$ in $L^1(\mathrm{\Omega })$ with all functions in $BV(\mathrm{\Omega })$, each compactly supported smooth unit test field has $\int u\mathrm{div}\varphi ={\lim }_k\int u_k\mathrm{div}\varphi \le {\mathrm{lim\hspace{0.17em}inf}}_{k}TV(u_k;\mathrm{\Omega })$. Taking the supremum using [F3] proves the lower-semicontinuity assertion. If only $u\in L^1$ is initially known and that liminf is finite, the Countable Choice characterization in [F3] also proves $u\in BV$.

Extend every $u_k$ by zero. By [F1], the functions $v_k={\tilde{u}}_k$ have support in the bounded set $\bar{\mathrm{\Omega }}$, satisfy ${\sup }_k\mathrm{\parallel }{v}_k{\mathrm{\parallel }}_1\le M$, and have $TV(v_k;{\mathbb{R}}^n)\le C_{\mathrm{\Omega }}M$. Their translation differences are at most $\mathrm{\mid }h\mathrm{\mid }{C}_{\mathrm{\Omega }}M$, uniformly in $k$. Hence [F2] gives a subsequence $v_{k_j}\to v$ in $L^1({\mathbb{R}}^n)$. Restriction yields $u_{k_j}\to u=v{\mathrm{\mid }}_{\mathrm{\Omega }}$ in $L^1(\mathrm{\Omega })$. The variations in $\mathrm{\Omega }$ are bounded by $M$, so step 1.1 and [F3] prove $u\in BV(\mathrm{\Omega })$ and $TV(u;\mathrm{\Omega })\le {\mathrm{lim\hspace{0.17em}inf}}_{j}TV(u_{k_j};\mathrm{\Omega })$. This proves the full compactness conclusion.