A combinatorial analog of Lyapunov’s theorem for infinitesimally generated atomic vector measures

It is shown that the range of a measure obtained by the addition of infinitesimal vectors is convex up to infinitesimal errors. By a well-known theorem of Lyapunov, the range in Rn of a nonatomic totally finite vector-measure is convex (see [5]). Recent developments in economics (see [3]) have established the need for a similar theorem for a purely atomic measure on a *finite set A, where the measure of each atom a E A is a vector of infinitesimal length. The desired result is a corollary of the following theorem of Steinitz ([7, pp. 167-172]; also see [1, pp. 148-149]): Given any finite collection of vectors in n-space Rn with sum 0 and maximum norm M, there is an ordering v1, v2, * * *, v1, of those vectors such that the norm of any partial sum 2=l vi is smaller than 2nM. (Note that the norm in Steinitz's theorem need not be the Euclidean norm, and if it is, 2nM may not be the best constant for dimension n.) (See [2] and [4].) Let *R and *N denote the nonstandard models for the real numbers R and natural numbers N in a fixed enlargement of a structure that contains R. (See [6].) We write a-b when a E *R, b E *R and a-b is infinitesimal, and we denote by lvl the Euclidean distance of any vector v from the origin 0. Fix n E *N, n finite or infinite. If u and v are vectors in *Rf, we shall write u-v if Iu-vJ2'O. Recall that a *finite set is a set for which there is an internal one-to-one correspondence with an initial segment of *N; such a set has all the "formal" properties of a finite set. THEOREM. Let A be a *finite set, and for each a E A, let v(a) be a vector in n-space *Rn with nv(a)-O. For each internal set Bc A, set Received by the editors April 3, 1972 and, in revised form, November 17, 1972. AMS (MOS) subject classifications (1970). Primary 26A98, 28A45, 46G10.