A bounded open Jordan set has an increasing exhaustion by compact finite unions of grid rectangles with vanishing content remainder

Enclose $V$ in a rectangle and choose nested dyadic grids whose meshes tend to zero. Let $K_j$ be the union of every closed cell of the $j$th grid that is contained in $V$. Only finitely many cells occur. Every child of a retained cell is retained, so $K_j\subseteq K_{j+1}$; each $K_j$ is compact, Jordan, and contained in $V$.

If compact $C\subseteq V$, the distance from $C$ to the closed complement of $V$ is positive. Once the mesh diameter is smaller than that distance, every grid cell meeting $C$ is contained in $V$, so $C\subseteq K_j$.

Every unretained cell meeting $V$ also meets a mesh-sized neighborhood of $\mathrm{\partial }V$. By [L1], that boundary has content zero; [L2] therefore makes the total volume of all such cells arbitrarily small for fine enough grids. Finite additivity [L3] bounds $\mathrm{cont}(V\setminus K_j)$ by that volume, proving the limit.