Integration over the levels of ACL-functions (Q2387872)

For a domain \(G\subset \mathbb{R}^n\) denote by ACL\(_1(G)\) the class of all functions that are continuous in \(G\) and belong to the Sobolev class \(W_{1,\text{loc}}^1(G)\). The following theorem is proved that sharpens corresponding results known in the theory of multidimensional variations: Let \(G\subset \mathbb{R}^n\) be a domain, \(M\subset G\) be a Lebesgue measurable set and \(r\in \text{ACL}_1(G)\). Then \[ \int_M | \nabla r(x)| \,dx =\int_{r(G)} H(\mathop{M}\limits_t)\,dt, \] where \(\mathop{M}\limits_t=\{x\in M: r(x)=t\}\) and \(H\) is the Hausdorff \((n-1)\)-measure. From the theorem it is derived a generalization of the classical formula of repeated integration.