On the Surfaces Moduli Theory

In this article we continue to develop the theory of several moduli of families of surfaces, in particular, strings (open surfaces) of various dimensions in Euclidean spaces. Since the surfaces in question can be extremely fractal (wild), the natural basis for studying them is the so-called Hausdorff measures. As is known, these moduli are the main geometric tool in the modern mapping theory and related topics in geometry, topology and the theory of partial differential equations with appropriate applications to the boundary-value problems of mathematical physics in anisotropic and inhomogeneous media. In addition, this theory can also find its further applications in many other fields, including mathematics itself (nonlinear dynamics, minimal surfaces), theoretical physics (conformal field theory, string theory), and engineering (mathematical models of the filtration of gases and fluids in underground mines of water, gas and oil seams, crystal growth and others). On the basis of the proof of Lemma 1 about the connections between moduli and the Lebesgue measures, we have proved the corresponding analogue of the Fubini theorem in the terms of the moduli that extends the known Väisälä theorem for families of curves to families of surfaces of arbitrary dimensions. It is necessary to note specially here that the most refined place in the proof of Lemma 1 is Proposition 1 on measurable (Borel) hulls of sets in Euclidean spaces. We also prove here the corresponding Lemma 2 and Proposition 2 on families of centered spheres. Finally, in a similar way, suitable results can be also obtained for families of several spheroids.