Editing Bounded variation (section)

===Locally BV functions===
If the [[function space]] of [[locally integrable function]]s, i.e. [[Function (mathematics)|function]]s belonging to <math>  L^1_\text{loc}(\Omega)</math>, is considered in the preceding definitions {{EquationNote|2|1.2}}, {{EquationNote|3|2.1}} and {{EquationNote|4|2.2}} instead of the one of [[integrable function|globally integrable functions]], then the function space defined is that of '''functions of locally bounded variation'''. Precisely, developing this idea for {{EquationNote|4|definition 2.2}}, a '''[[local property|local]] variation''' is defined as follows,

: <math> V(u,U):=\sup\left\{\int_\Omega u(x)\operatorname{div}\boldsymbol{\phi}(x) \, \mathrm{d}x : \boldsymbol{\phi} \in C_c^1(U,\mathbb{R}^n),\ \Vert\boldsymbol{\phi}\Vert_{L^\infty(\Omega)}\le 1\right\}</math>

for every [[Set (mathematics)|set]] <math>  U\in\mathcal{O}_c(\Omega)</math>, having defined <math>  \mathcal{O}_c(\Omega)</math> as the set of all [[Relatively compact subspace|precompact]] [[open subset]]s of '''<math>\Omega</math>''' with respect to the standard [[topology]] of [[dimension (mathematics)|finite-dimensional]] [[vector space]]s, and correspondingly the class of functions of locally bounded variation is defined as
:<math>\operatorname{BV}_\text{loc}(\Omega)=\{ u\in L^1_\text{loc}(\Omega)\colon \, (\forall U\in\mathcal{O}_c(\Omega)) \, V(u,U)<+\infty\}</math>