Bounded BV data on the real line

Let $u$ be an almost-everywhere equivalence class of measurable functions on $\mathbb{R}$ with $u\in L^{\mathrm{\infty }}(\mathbb{R})\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathbb{R})$, using the quotient convention of The space $L^p(\mu )$ as the quotient by null functions, the essentially bounded class of The space $L^{\mathrm{\infty }}(\mu )$ of essentially bounded measurable functions, and the local integrability convention of A locally integrable function on ${\mathbb{R}}^n$. For $h>0$ define $D_{h}u(x):=u(x+h)-u(x),{\mathrm{TV}}_{\mathrm{t}\mathrm{r}}(u):={\sup }_{h>0}\frac{1}{h}{\int }_{\mathbb{R}}\mathrm{\mid }{D}_{h}u(x)\mathrm{\mid }dx,$ where the integral may be $+\mathrm{\infty }$. The bounded BV class on the line ${\mathrm{B}\mathrm{V}}_b(\mathbb{R})$ consists of those classes for which ${\mathrm{TV}}_{\mathrm{t}\mathrm{r}}(u)<\mathrm{\infty }$; we write $\mathrm{TV}(u):={\mathrm{TV}}_{\mathrm{t}\mathrm{r}}(u)$.

This translation-variation convention is the standard one-dimensional BV seminorm on bounded locally integrable classes. It is unchanged by modifying $u$ on a null set, because each translate of a null set is null. It does not require $u\in L^1(\mathbb{R})$: every bounded constant function belongs to ${\mathrm{B}\mathrm{V}}_b(\mathbb{R})$ and has variation $0$.