 Bounded-variation contraction under clipping

Let $T(t)=\min (b,\max (a,t))$, $a\le b$, and let $u\in BV(\Omega )$. The variation contractionfollows from the one-Lipschitz scalar composition rule, by smooth approximation and lower semicontinuity. To conclude global membership $T\circ u\in BV(\Omega )$, one must also retain its L1 norm.

That integrability is automatic when $\Omega$ has finite Lebesgue measure. It also holds on an arbitrary domain when $T(0)=0$, equivalently $0\in [a,b]$, since $|T(u)|\le |u|$. Without either condition it can fail: $u=0$ on $\mathbb{R}$ belongs to BV space, but clipping it to $[1,2]$ gives the nonintegrable constant one. Its derivative variation is zero, which does not repair the missing global integrability.

For total variation denoising on a bounded domain with data in $[a,b]$, clipping cannot increase either the variation penalty or the pointwise squared error. Uniqueness of the minimizer therefore forces range preservation. The variation inequality and this integrability qualification describe different parts of the argument.