The third set has measure zero on which every element has a unique structural representation that satisfies the restrictions.

On the other hand, the set of structural parameters that are globally identified is of measure zero because the constraint [b.sup.2.sub.1] - [4d.sub.1] = [b.sup.2.sub.2] - [4d.sub.2] = 0 must be met.

The set G, defined by (9), contains all such locally unidentified points but has measure zero according to Theorems 6 and 8.

In this appendix, we highlight a few properties of differentiable manifolds and develop some results about measure zero subsets of differentiable manifolds.

We exploit this local Euclidean structure to characterizer sets of measure zero on manifolds.

While measure zero sets are often studied in the context of Lebesgue measure, for Euclidean spaces we do not need the full power of this machinery.

A set A [subset] [R.sub.k] is of measure zero if and only if for every [epsilon] > 0 there exist countably many closed k-dimensional rectangles [R.sub.i] of volume [v.sub.i] such that A [subset] [U.sub.i] [R.sub.i] and [[SIGMA].sub.i] [v.sub.i] < [epsilon].

This definition of measure zero is equivalent to definition arising from Lebesgue measure.

The set A is of measure zero in M if and only if for every coordinate coordinate system (V, h), the set ([pi] o h) (A [intersection] V) is of measure zero in [R.sup.k].

It must be the case that for any A [subset] M the set ([pi] o [h.sub.1]) (A [intersection] [V.sub.1] [intersection] [V.sub.2]) is of measure zero if and only if ([pi] o [h.sub.2]) (A [intersection] [V.sub.1] [intersection] [V.sub.2]) is of measure zero.

The first-order condition 10 implicitly defines a function [R.sub.i] = [PHI]([R.sub.j]), except possibly at a point of measure zero.

Moreover, because the theorem fails only at a point of measure zero, the function certainly exists around neighborhoods at all other points.