Approximation of the Volume

We deal here with a basic question: consider a smooth (compact oriented) submanifold ^Mn of the Euclidean space E^N, n < N, and a (measurable) subset W, “close to ^Mn.” Can we approximate the volume Vol _n (^Mn) of ^Mn by Vol _n (W)? We have seen in Sect. 3.1.3 that the well-known “Lantern of Schwarz” shows that the area of a sequence of triangulations inscribed in a fixed cylinder of E³ may tend to infinity when the sequence tends to the cylinder for the Hausdorff topology. We give here a general approximation theorem by adding a suitable geometric assumption: we assume that the tangent bundle of the sequence tends to the tangent bundle of ^Mn, in precise sense.