Lusin approximation for functions of bounded variation

Abstract

We prove a Lusin approximation of functions of bounded variation. If $f$ is a function of bounded variation on an open set $\Omega\subset X$, where $X=(X,d,\mu)$ is a given complete doubling metric measure space supporting a $1$-Poincar\'e inequality, then for every $\varepsilon>0$, there exist a function $f_\varepsilon$ on $\Omega$ and an open set $U_\varepsilon\subset\Omega$ such that the following properties hold true: \begin{enumerate} \item ${\rm Cap}_1(U_\varepsilon)<\varepsilon$; \item $\|f-f_\varepsilon\|_{\BV(\Omega)}< \varepsilon$; \item $f^\vee\equiv f_\varepsilon^\vee$ and $f^\wedge\equiv f_\varepsilon^\wedge$ on $\Omega\setminus U_\varepsilon$; \item $f_\varepsilon^\vee$ is upper semicontinuous on $\Omega$, and $f_\varepsilon^\wedge$ is lower semicontinuous on $\Omega$. \end{enumerate} If the space $X$ is unbounded, then such an approximating function $f_\varepsilon$ can be constructed with the additional property that the uniform limit at infinity of both $f^\vee_\varepsilon$ and $f^\wedge_\varepsilon$ is $0$. Moreover, when $X=\R^d$, we show that the non-centered maximal function of $f_\varepsilon$ is continuous in $\Omega$.