Variation of the dyadic maximal function

We prove that for the dyadic maximal operator $\mathrm{M}$ and every locally integrable function $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^d)$ with bounded variation, also $\mathrm{M}f$ is locally integrable and $\mathrm{M}f\le C_{d}f$ for any dimension $d\ge 1$. It means that if $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^d)$ is a function whose gradient is a finite measure then so is $\mathrm{\nabla }\mathrm{M}f$ and $|\nabla \mathrm M f|{L^1(\mathbb R^d)}\leq C_d|\nabla f|{L^1(\mathbb R^d)}$. We also prove this for the local dyadic maximal operator.