## The multidimensional fundamental theorem of calculus.(English)Zbl 0638.26011

If $$E\subset R^ m$$, $$\eta >0$$ and $$U(E,\eta)=\{x\in R^ m:$$ $$dist(x,E)<\eta \}$$, E is said to be thin (resp. slight) if it is compact and $$U(E,\eta)=O(\eta)$$ (resp. $$=o(\eta))$$ as $$\eta\to 0$$. A bounded subset of $$R^ m$$ whose boundary is thin is called admissible. The present paper proposes a Riemann type integration for functions defined on admissible sets of $$R^ n$$, which coincides with Lebesgue integral for positive functions, provides a Stokes theorem for differentiable vector fields and admits a transformation theorem.
Reviewer: J.Mawhin

### MSC:

 26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.) 26B15 Integration of real functions of several variables: length, area, volume 26B30 Absolutely continuous real functions of several variables, functions of bounded variation 58C35 Integration on manifolds; measures on manifolds