 Failure of L2 closure of the global BV domain

For $1<a\le 2$, the radial function $f(x)=(1+|x|{)}^{-a}$ belongs to $L^2({\mathbb{R}}^2)$ and has finite distributional total variation seminorm on a domain, but is not in $L^1({\mathbb{R}}^2)$. Cutting it off outside radius $T$ gives bounded-variation space functions tending to $f$ in $L^2$, with variation tending to that of $f$: the added jump costs $2\pi T(1+T{)}^{-a}\to 0$. Thus the penalty equal to TV on $BV\cap L^2$ and infinity elsewhere is not sequentially lower semicontinuous in $L^2$. The closed homogeneous bounded-variation space extension avoids this domain issue.